Properties of Normal to a Parabola

Co-normal Points

Edited By Komal Miglani | Updated on Jul 02, 2025 07:45 PM IST

The line perpendicular to the tangent to the curve at the point of contact is normal to the parabola. A maximum of three normals can be drawn from a point to a parabola and their feet (points) where they meet the parabola are called the co-normal points. In real life, we use normal to determine the slope and curvature of roads and bridges.

This Story also Contains

What is Co-normal Points?
Proof of Co-normal Pointsṁṁ
Pair of Tangent of a Parabola
Derivation of Pair of Tangent of a Parabola
Properties of co-normal points
Problems Based on Co-normal Points

Co-normal Points

In this article, we will cover the concept of the Co-normal Points. This category falls under the broader category of Coordinate Geometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. A total of twenty questions have been asked on JEE MAINS( 2013 to 2023) from this topic including one in 2022.

What is Co-normal Points?

The points on the curve at which the normals pass through a common point are called co-normal points. In general, a maximum of three normals can be drawn from a point to a parabola and their feet (points) where they meet the parabola are called the co-normal points.

Proof of Co-normal Pointsṁṁ

P (h,k) is any point on the plane

equation of normal to the parabola $y^2=4 a x$ is

$
\mathrm{y}=\mathrm{mx}-2 \mathrm{am}-\mathrm{am}^3
$
It passes through the point $\mathrm{P}(\mathrm{h}, \mathrm{k})$ then,

$
\begin{aligned}
& \mathrm{k}=\mathrm{mh}-2 \mathrm{am}-\mathrm{am}^3 \\
& \text { or, } \mathrm{am}^3+\mathrm{m}(2 \mathrm{a}-\mathrm{h})+\mathrm{k}=0
\end{aligned}
$
This cubic equation have three roots say, $m_1, m_2, m_3$

$
\begin{aligned}
& m_1+m_2+m_3=0 \\
& m_1 m_2+m_2 m_3+m_3 m_1=\frac{2 a-h}{a} \\
& m_1 m_2 m_3=-\frac{k}{a}
\end{aligned}
$

Points A, B, and C in which the three normals from P (h,k) meet the parabola are called co-normal points.

Pair of Tangent of a Parabola

The combined equation of the pair of tangents drawn from an external point P(x₁,y₁) to the parabola, say S=y²-4ax=0 is SS₁=T².

Where,

$\begin{aligned} & \mathrm{S}=\mathrm{y}^2-4 \mathrm{ax} \\ & \mathrm{S}_1=\mathrm{y}_1^2-4 \mathrm{ax}_1 \\ & \mathrm{~T}=\mathrm{yy}_1-2 \mathrm{a}\left(\mathrm{x}+\mathrm{x}_1\right) \\ & \left(\mathrm{y}^2-4 \mathrm{ax}\right)\left(\mathrm{y}_1^2-4 \mathrm{ax}_1\right)=\left(\mathrm{yy}_1-2 \mathrm{a}\left(\mathrm{x}+\mathrm{x}_1\right)\right)^2\end{aligned}$

Derivation of Pair of Tangent of a Parabola

Let (h, k) be any point on either of the tangents drawn from the point P (x₁, y₁). The equation of the line joining the point (x₁, y₁) and (h, k) is

$
\begin{aligned}
\mathrm{y}-\mathrm{y}_1 & =\frac{\mathrm{k}-\mathrm{y}_1}{\mathrm{~h}-\mathrm{x}_1}\left(\mathrm{x}-\mathrm{x}_1\right) \\
\Rightarrow \quad \mathrm{y} & =\frac{\mathrm{k}-\mathrm{y}_1}{\mathrm{~h}-\mathrm{x}_1} \mathrm{x}+\frac{\mathrm{hy}_1-\mathrm{kx}_1}{\mathrm{~h}-\mathrm{x}_1}
\end{aligned}
$
For this to be a tangent to the parabola it must be of the form $y=m x+\frac{a}{m}$ comparing this two equations, we get

$
m=\frac{k-y_1}{h-x_1} \text { and } \frac{a}{m}=\frac{h y_1-k x_1}{h-x_1}
$
Therefore, by multiplication we get

$
\begin{aligned}
\quad & a=\left(\frac{k-y_1}{h-x_1}\right)\left(\frac{h y_1-k x_1}{h-x_1}\right) \\
\Rightarrow \quad & a\left(h-x_1\right)^2=\left(k-y_1\right)\left(h y_1-k x_1\right)
\end{aligned}
$

Note:

The formula SS₁ = T² works for finding a pair of tangents to any general parabola as well.

Properties of co-normal points

1) The sum of the ordinates of the feet of conormal points is zero.

2) Circle through the conormal points passes through the origin (vertex of the parabola).

3) The Centroid of the triangle formed by co-normal points lies on the x-axis or on the axis of the parabola in general.

4) Three normals can be drawn from a point to a parabola.

5) Three normals can be drawn from a point to a parabola.

Problems Based on Co-normal Points

Example 1: If the normals drawn to the parabola, $y^2=2 x$ pass through the point $(a, 0) a \neq 0$, then 'a' must be greater than :
[JEE MAINS 2021]
Solution: For more than 3 normals (on axis)
$x>\frac{L}{2} \quad$ (where $L$ is length of L.R.)
For $y^2=2 x$
L. $R .=2$

For $(a, 0)$

$
a>\frac{\text { L.R. }}{2} \Rightarrow a>1
$

Hence, the answer is 1

Example 2: Find the locus of a point from where a normal can be drawn to the parabola which makes an angle of 45^oin an anticlockwise direction with the +x-axis

Solution: Let point be P(h,k) and let m₁, m₂, m₃ be slopes of normals drawn from this point

Thus m₁= tan 45^o=1

Now put the value of m₁=1 in relations derived for co-normal points

$\begin{aligned} & m_1+m_2+m_3=0 \Rightarrow m_2+m_3=-1 \\ & m_1 m_2+m_2 m_3+m_3 m_1=\frac{2 a-h}{a} \Rightarrow m_2+m_2 m_3+m_3=\frac{2 a-h}{a} \\ & m_1 m_2 m_3=-\frac{k}{a} \Rightarrow m_2 m_3=-\frac{k}{a}\end{aligned}$

Solving the above three equation
$
\frac{\mathrm{h}-2 \mathrm{a}}{a}=1+\frac{k}{a} h-k=3 a
$

or $x-y-3 a=0$

Hence, the answer is x-y-3a =0

Example 3: If three distinct and real normals can be drawn to $y^2=8 \mathrm{x}$ from the point $(\mathrm{a}, 0)$, then

Solution
The equation of normal in terms of $m$ is

$
y=m x-4 m-2 m^3
$
If it passes through $(\mathrm{a}, 0)$, then

$
\begin{aligned}
& \mathrm{am}-4 \mathrm{~m}-2 \mathrm{~m}^3=0 \\
& \Rightarrow \mathrm{m}\left(\mathrm{a}-4-2 \mathrm{~m}^2\right)=0 \\
& \Rightarrow \mathrm{m}=0, \mathrm{~m}^2=\frac{\mathrm{a}-4}{2}
\end{aligned}
$
For three distinct normals,

$
\begin{aligned}
& a-4>0 \\
& a>4
\end{aligned}
$

Hence, the answer is a>4

Example 4: If the normals drawn from a point to the parabola $x^2=4 y$ cut the line $y=2$ in points whose abscissa are in A.P., then the slopes of the tangents at the three co-normal points are in

Solution: Any point on $\mathrm{x}^2=4 y_{\text {is }}\left(2 \mathrm{t}, \mathrm{t}^2\right)$
Normal at this point is $y-t^2=-\frac{1}{t}(x-2 t)$
Or $\mathrm{x}+\mathrm{ty}=2 \mathrm{t}+\mathrm{t}^3$
It is drawn from $\mathrm{A}\left(\mathrm{x}_1, \mathrm{y}_1\right)$. The condition that is

$
\mathrm{x}_1+\mathrm{ty}_1=2 \mathrm{t}+\mathrm{t}^3
$
Or $\mathrm{t}^3+\mathrm{t}\left(2-\mathrm{y}_1\right)-\mathrm{x}_1=0$
Let its roots by $\mathrm{t}_1, \mathrm{t}_2, \mathrm{t}_3$

$
\therefore \quad \mathrm{t}_1+\mathrm{t}_2+\mathrm{t}_3=0
$
The intersection of (1) with $\mathrm{y}=2$ is given by

$
\begin{aligned}
& x+2 t=2 t+t^3 \\
& t^3=x
\end{aligned}
$
The abscissa of three points of intersection are

$
\mathrm{t}_1^3, \mathrm{t}_2^3, \mathrm{t}_3^3
$

They are in A.P.

$
\Rightarrow \quad 2 \mathrm{t}_2^3=\mathrm{t}_1^3+\mathrm{t}_3^3
$
Or $3 \mathrm{t}_2^3=\mathrm{t}_1^3+\mathrm{t}_2^3+\mathrm{t}_3^2$
$A s \mathrm{t}_1+\mathrm{t}_2+\mathrm{t}_3=0 \Rightarrow\left(\mathrm{t}_1+\mathrm{t}_2+\mathrm{t}_3\right)^3=0$
Or $\mathrm{t}_1^3+\mathrm{t}_2^3+\mathrm{t}_3^3=3 \mathrm{t}_1 \mathrm{t}_2 \mathrm{t}_3$
From (3) and (4)
$3 \mathrm{t}_2^3=3 \mathrm{t}_1 \mathrm{t}_2 \mathrm{t}_3$
Or $\mathrm{t}_2^2=\mathrm{t}_1 \mathrm{t}_3$
$\frac{1}{t_1} \frac{1}{t_3}=\frac{1}{t_2^2}$
But $\frac{1}{t_1}, \frac{1}{t_2}, \frac{1}{\mathrm{t}_3}$ are slopes of tangents at three conormal points.
Therefore, they are in G.P.

Example 5: Tangents PA and PB are drawn to parabola $\mathrm{y}^2=4 \mathrm{ax}$. If the slope of the bisector of the angle PAB is $\sqrt{3}$, then the locus of the point ' P ' is

Solution: The slope of tangents will be

$
\tan \left(\frac{\pi}{3}+\theta\right), \tan \left(\frac{\pi}{3}-\theta\right)
$

$
\begin{aligned}
& \text { Let } \mathrm{A} \equiv\left(\mathrm{a}^{\mathrm{t}^2}, 2 \mathrm{a}^{\mathrm{t}_1}\right), \mathrm{B} \equiv\left(\mathrm{a}^{\mathrm{t}_2^2}, 2 \mathrm{at}_2\right) \\
& \Rightarrow \quad \mathrm{P} \equiv\left[\mathrm{at}_1 \mathrm{t}_2, \mathrm{a}\left(\mathrm{t}_1+\mathrm{t}_2\right)\right]
\end{aligned}
$
We have

$
\begin{aligned}
& \mathrm{t}_1=\cot \left(\frac{\pi}{3}+\theta\right), \mathrm{t}_2=\cot \left(\frac{\pi}{3}-\theta\right) \\
& \Rightarrow \mathrm{t}_1=\frac{\frac{1}{\sqrt{3}} \cdot \cot \theta-1}{\frac{1}{\sqrt{3}}+\cot \theta}, \mathrm{t}_2=\frac{\frac{1}{\sqrt{3}} \cdot \cot \theta+1}{\frac{1}{\sqrt{3}}-\cot \theta} \\
& \Rightarrow \quad \cot \theta=\frac{\sqrt{3}+\mathrm{t}_1}{1-\sqrt{3} \mathrm{t}_1}=\frac{\mathrm{t}_2-\sqrt{3}}{1+\sqrt{3} \mathrm{t}_2} \\
& \Rightarrow \quad \sqrt{3}+3 \mathrm{t}_2+\mathrm{t}_1+\sqrt{3} \mathrm{t}_1 \mathrm{t}_2=\mathrm{t}_2-\sqrt{3}-\sqrt{3} \mathrm{t}_1 \mathrm{t}_2+3 \mathrm{t}_1 \\
& \Rightarrow \quad 2 \sqrt{3}+2 \sqrt{3} \mathrm{t}_1 \mathrm{t}_2=2\left(\mathrm{t}_1-\mathrm{t}_2\right) \\
& \Rightarrow \quad 3\left(1+\mathrm{t}_2 \mathrm{t}_2\right)^2=\left(\mathrm{t}_1-\mathrm{t}_2\right)^2=\left(\mathrm{t}_1+\mathrm{t}_2\right)^2-4 \mathrm{t}_1 \mathrm{t}_2
\end{aligned}
$
Thus, the locus of $p$ is

$
y^2=3 x^2+3 a^2+10 a x
$

Hence, the answer is $y^2=3 x^2+3 a^2+10 a x$

Frequently Asked Questions (FAQs)

1. Can co-normal points be used to construct tangent lines to a conic section?

Yes, co-normal points can be used to construct tangent lines. Since the normal at a point is perpendicular to the tangent at that point, knowing the normal line at a co-normal point allows you to easily construct the tangent line at that point.

2. How do co-normal points behave as the eccentricity of a conic section approaches 1?

As the eccentricity approaches 1, the conic section transitions from an ellipse to a parabola. The behavior of co-normal points changes from finite pairs in ellipses to infinitely many pairs in parabolas, reflecting the changing geometry of the curve.

3. Can co-normal points be used to determine the type of conic section?

While co-normal points alone cannot definitively determine the type of conic section, their behavior and distribution can provide clues. For example, the existence of infinitely many pairs of co-normal points suggests a parabola, while a finite number of pairs indicates an ellipse or hyperbola.

4. How do you find the equation of the normal at a point on a conic section?

To find the equation of the normal at a point (x₀, y₀) on a conic section, first find the slope of the tangent line at that point using implicit differentiation. The normal line will be perpendicular to this tangent, so its slope will be the negative reciprocal of the tangent's slope.

5. How do co-normal points relate to the concept of pedal curves of conic sections?

The pedal curve of a conic section is the locus of the feet of perpendiculars drawn from a fixed point to the tangents of the conic. Co-normal points can help in understanding pedal curves, as they provide information about normals and tangents at specific points on the conic.

6. How do co-normal points relate to the concept of orthoptic curves of conic sections?

The orthoptic curve of a conic section is the locus of points from which two tangents to the conic are perpendicular. While not directly equivalent, understanding co-normal points can provide insights into the geometric relationships that give rise to orthoptic curves.

7. How do co-normal points behave in degenerate conic sections?

In degenerate conic sections (such as a point, line, or pair of lines), the concept of co-normal points may not apply in the traditional sense. However, understanding co-normal points can help in analyzing how these degenerate cases arise as limiting cases of regular conic sections.

8. How can the concept of co-normal points be extended to three-dimensional quadric surfaces?

The concept of co-normal points can be extended to three-dimensional quadric surfaces (like ellipsoids or hyperboloids) by considering pairs of points where the normal planes intersect along a line that lies on a principal plane of the surface. This extension helps in understanding the geometry and symmetry of these more complex surfaces.

9. Can co-normal points be used to study the projective properties of conic sections?

Yes, co-normal points can be useful in studying projective properties of conic sections. Their invariant relationships under certain transformations can provide insights into how conic sections behave in projective geometry.

10. How do co-normal points help in understanding the concept of confocal conic sections?

While co-normal points are not directly related to confocal conics, understanding their properties can aid in grasping the geometric relationships in confocal systems. Both concepts involve special points and lines that highlight the symmetric and focal properties of conic sections.

11. What is the significance of co-normal points in ellipses?

In ellipses, co-normal points help us understand the relationship between the curve and its axes. They provide insight into the symmetry of the ellipse and can be used to solve problems related to tangents and normals of the curve.

12. How do you find co-normal points on a parabola?

To find co-normal points on a parabola, you need to solve a system of equations involving the parabola's equation and the equations of the normals at two points. The solution will give you the coordinates of the co-normal points that satisfy these conditions.

13. Can a parabola have infinitely many pairs of co-normal points?

Yes, a parabola can have infinitely many pairs of co-normal points. This is because for any point on a parabola, there exists another point (its co-normal partner) such that their normals intersect on the axis of the parabola.

14. What is the role of co-normal points in understanding the curvature of conic sections?

Co-normal points provide insight into the curvature of conic sections. The distance between co-normal points and their corresponding intersection point on the axis is related to the radius of curvature at those points, helping us understand how the curve bends.

15. How do co-normal points relate to the eccentricity of a conic section?

The eccentricity of a conic section affects the distribution of co-normal points. In ellipses (e < 1), co-normal points are finite in number. In parabolas (e = 1), there are infinitely many pairs. In hyperbolas (e > 1), the distribution is more complex and depends on the specific hyperbola.

16. What is the relationship between co-normal points and the center of an ellipse?

In an ellipse, the center is the midpoint of the line segment joining any pair of co-normal points. This property highlights the symmetry of co-normal points with respect to the center of the ellipse.

17. What is the geometric interpretation of co-normal points on a hyperbola?

On a hyperbola, co-normal points are pairs of points where the normals to the curve intersect on one of the axes (transverse or conjugate). They help in understanding the symmetry and curvature properties of the hyperbola.

18. How do co-normal points differ between ellipses and hyperbolas?

In ellipses, co-normal points always exist in pairs on the same branch of the curve. In hyperbolas, co-normal points can exist on the same branch or on different branches, depending on the specific hyperbola and its eccentricity.

19. Can co-normal points coincide?

No, co-normal points cannot coincide. By definition, they are distinct points on the conic section where the normals intersect at a point on the axis. If they were to coincide, it would violate the fundamental property of co-normal points.

20. How are co-normal points related to the axis of a conic section?

Co-normal points always occur in pairs, and the point where their normals intersect lies on the axis of the conic section. This intersection point on the axis is equidistant from the two co-normal points, creating a symmetrical relationship between the points and the axis.

21. What are co-normal points in conic sections?

Co-normal points are pairs of points on a conic section where the normals (perpendicular lines) to the curve at these points intersect at a point on the axis of the conic. These points have a special geometric relationship and play an important role in understanding the properties of conic sections.

22. Can a circle have co-normal points?

Yes, a circle can have co-normal points. In fact, any two points on a circle that are on opposite ends of a diameter are co-normal points. Their normals (which are radii in this case) intersect at the center of the circle, which lies on all axes of symmetry.

23. What is the significance of co-normal points in the study of conics using polar coordinates?

In polar coordinates, co-normal points can be used to understand the radial and angular relationships within conic sections. They provide insight into the symmetry and geometric properties of conics when expressed in polar form, particularly in relation to the focus and directrix.

24. How are co-normal points related to the concept of conjugate diameters in ellipses?

Co-normal points and conjugate diameters are both related to the symmetry properties of ellipses. While not directly equivalent, understanding co-normal points can aid in grasping the concept of conjugate diameters, as both involve pairs of points with special geometric relationships.

25. What is the role of co-normal points in understanding the osculating circle of a conic section?

Co-normal points can aid in understanding osculating circles, which are circles that best approximate the curvature of a conic section at a given point. The radius of curvature at co-normal points, which is related to their distance from the axis, helps in constructing and analyzing these osculating circles.

26. What is the relationship between co-normal points and the latus rectum of a conic section?

The latus rectum is a chord of the conic section that passes through a focus and is perpendicular to the major axis. While not directly related, understanding co-normal points can help in problems involving the latus rectum, as both concepts relate to the geometry and properties of the conic section.

27. What is the significance of co-normal points in optical applications of conic sections?

In optics, co-normal points are important for understanding reflection properties of conic sections. For example, in elliptical mirrors, light rays originating from one focus and reflecting off co-normal points will converge at the other focus.

28. Can co-normal points be used to solve optimization problems involving conic sections?

Yes, co-normal points can be useful in certain optimization problems involving conic sections. For example, they can help in finding points on the conic where specific geometric conditions are met, which is often a key step in optimization problems.

29. What is the relationship between co-normal points and the axes of symmetry in conic sections?

Co-normal points always have their normal lines intersecting on an axis of symmetry of the conic section. This property reinforces the fundamental role of symmetry in the geometry of conic sections and helps in understanding their overall structure.

30. What is the significance of co-normal points in the study of caustics of conic sections?

Caustics are envelope curves formed by reflected or refracted rays from a curved surface. Co-normal points can be useful in studying caustics of conic sections, as they provide information about normal lines which are crucial in determining the behavior of reflected or refracted rays.

31. How can co-normal points be used to understand the concept of polar reciprocation in conic sections?

Co-normal points can provide insights into polar reciprocation, a concept in projective geometry. The relationship between co-normal points and their normals' intersection on the axis can be related to the polar-polar relationship, helping to illustrate this more advanced concept.

32. What is the relationship between co-normal points and the concept of conjugate points in conic sections?

While co-normal points and conjugate points are distinct concepts, they both relate to special geometric relationships within conic sections. Understanding co-normal points can provide a foundation for grasping the more complex idea of conjugate points, as both involve pairs of points with specific properties.

33. How do co-normal points behave in the limiting case when a conic section approaches a straight line?

As a conic section approaches a straight line (a degenerate case), the behavior of co-normal points becomes less distinct. In the limit, the concept loses its traditional meaning, but understanding this limiting behavior can provide insights into the continuity of geometric properties in conic sections.

34. Can the concept of co-normal points be applied to higher-order curves beyond conic sections?

While co-normal points are specifically defined for conic sections, similar concepts can be extended to higher-order curves. For more complex curves, points where normals intersect in specific ways can provide insights into the curve's geometry, symmetry, and curvature properties.

35. How do co-normal points relate to the concept of focal chords in conic sections?

Focal chords are lines passing through a focus of a conic section and terminating at the curve. While not directly related, understanding co-normal points can enhance the comprehension of focal properties, including focal chords, as both concepts involve special geometric relationships within the conic.

36. What is the significance of co-normal points in understanding the evolute of a parabola?

The evolute of a parabola is a semicubical parabola. Co-normal points can help in understanding this evolute, as the points where normals of co-normal points intersect the axis of the parabola form part of this evolute curve.

37. How can co-normal points be used to illustrate the principle of duality in projective geometry of conics?

Co-normal points can serve as a concrete example to illustrate the principle of duality in projective geometry. The relationship between points on a conic and their normals intersecting on a line (the axis) can be viewed as a dual relationship, helping to explain this abstract concept.

38. What is the role of co-normal points in understanding the metric properties of conic sections?

Co-normal points play a significant role in understanding metric properties of conic sections. They provide information about distances, angles, and curvature, which are fundamental to the metric study of these curves.

39. How do co-normal points relate to the concept of asymptotes in hyperbolas?

While co-normal points and asymptotes are distinct concepts, understanding co-normal points can aid in grasping the behavior of hyperbolas near their asymptotes. The distribution of co-normal points changes as you move along the hyperbola, providing insights into its overall shape and asymptotic behavior.

40. Can co-normal points be used to study the properties of conic sections in non-Euclidean geometries?

The concept of co-normal points can be extended to study conic sections in non-Euclidean geometries. While the specific properties may change, the general idea of points with special normal relationships can provide insights into the behavior of curves in these alternative geometric systems.

41. How do co-normal points help in understanding the concept of reciprocal polars of conic sections?

Co-normal points can aid in understanding reciprocal polars by providing concrete examples of how points and lines are related in a conic section. This can serve as a stepping stone to grasping the more abstract concept of reciprocal polars in projective geometry.

42. What is the significance of co-normal points in the study of conic sections as projections of a circle?

Co-normal points can help illustrate how properties of a circle are transformed when projected to form other conic sections. Understanding how co-normal points behave under these projections can provide insights into the relationships between different types of conic sections.

43. How can co-normal points be used to explore the concept of inversion with respect to a circle?

While not directly related to inversion, understanding co-normal points can provide a foundation for exploring more complex transformations like inversion. The geometric relationships highlighted by co-normal points can serve as a basis for understanding how points and lines behave under inversion.

44. What is the role of co-normal points in understanding the focal properties of conic sections?

Co-normal points, while not directly related to foci, can enhance understanding of focal properties. They provide additional geometric relationships within the conic section that complement the focal definitions, helping to build a more comprehensive view of these curves.