Equation of Tangents to an Ellipse, Point of Contact, Practice Problems, FAQs

A line can either intersect an ellipse at two distinct points or touch it at a point or can pass without touching or intersecting it. When a line intersects an ellipse at distinct points, it is called an intersecting line. When the line touches the ellipse, then it is called a tangent. When the line neither cuts nor touches the ellipse, then it will be termed as a non-intersecting line. Look at the figure below to visualize these cases.

We have certain conditions associated with each situation shown in the figure above. Let’s try to understand each one of them in detail.

Table of Contents

• A Line and an Ellipse
• Equation of Tangents to an Ellipse
• Practice Problems
• FAQs

A Line and an Ellipse

Let us consider the line, (), and the ellipse,(),

To find the different conditions for different situations, we will solve both the equations simultaneously.

By substituting the value of from the equation of line in the equation of ellipse, we get the following equation:

or

Equation is quadratic in .

Now, discriminant,

Now,

Now, we will take different cases to solve for different conditions

Case 1 : The line will intersect the ellipse in two distinct points. In this case the line is secant to the ellipse. Case 2 : The line touches the ellipse or the line is tangent to the ellipse. Hence, if line is a tangent to ellipse     Then, Equation of tangent will be: and Also, these two equations represent parallel tangents to the ellipse. Case 3 : The line does not meet the ellipse.

Equation of Tangent to an Ellipse

The equation of tangent to an ellipse can be written in the following three ways :

(a) Point form: 

Firstly convert the given equation of ellipse in form by bringing everything to the left side of equality.

So, for standard equation of ellipse

Now, the equation of tangent to ellipse will be given by

To get , we replace , if we are finding an equation of tangent at point

⇒

(b) Parametric form:

We know that the parametric coordinates on an ellipse for any general point P are given by

To get an equation of tangent to an ellipse in parametric form, we can directly substitute these coordinates in Point Form discussed above i.e.

On substituting coordinates of P we get:

Hence the equation of tangent in parametric form will be:

(c) Slope form: 

When we were discussing different cases of a line and and ellipse, then we have solved 

an ellipse given by and line .

There we have obtained the value of as

Hence, The equation of tangents of slope to the given ellipse will be a

Point of contact (POC)/Point of tangency (POT) :

We know that the equations of tangent at points and ) are given by: 

where

As we know that both the equations represent the same line, the ratio of their coefficients will be equal.

Now, after solving, we get the following and

So, the point of tangency or point of contact is

Now, as we know that has two values, we will get the two respective coordinates for

Practice Problems

Example: If the tangents of the ellipse at the points and are perpendicular to each other, then what is the value of ?

Answer:

Step 1 : Given, equation of the ellipse, Firstly we have to rewrite this in standard form.  ⇒ ⇒ (Standard vertical ellipse) Also, and are perpendicular to each other. Let us consider the slope of ⇒ Now, by using the point form of a tangent to an ellipse, we get the following: ⇒ ⇒ Step 2 : As we can see that and ⇒ ⇒ Also, point (p, q) lies on tangent . ⇒ Step 3 : Now, by solving equations , we get the following: ⇒ ⇒ Hence, option (d) is the correct answer.

Example : Find the equation of tangent(s) to the ellipse whose inclination is . Also, find the point of tangency.

Solution : 

Step 1 : Given, equation of ellipse ⇒ ⇒ (Standard horizontal ellipse) Also, inclination of a tangent is i.e. ⇒ If line is a tangent to ellipse , then Step 2 : We get the following: Thus, on solving the equation of tangents, we get the following: Step 3 : Using, Line is a tangent to ellipse , then the point of tangency or point of contact is as follows: When  , then When , then

Example : If tangents are drawn to ellipse at all the points on the ellipse other than its four vertices, then the midpoints of the tangents intercepted between the coordinate axes lie on which curve?

Answer:

Step 1 : Given, equation of ellipse   ⇒ ⇒ (Standard horizontal ellipse) Let us consider to be any point on the ellipse (other than the vertices). And a point , i.e., the midpoint of the segment of tangent between the axes. As we know, ≡ Step 2 : Now, Equation of a tangent at point by using parametric form,  Now, coordinates of point And Here, by using the midpoint formula, we get the following: and Step 3 : and By adding both the terms after squaring, we get the following: Now, by replacing by and by , we get the following: Hence, option is the correct answer.

Example : Find the points on ellipse such that the tangent at each point makes equal angles with the axes.

Answer:

FAQs

Q 1.How many parallel tangents can be drawn to an ellipse?
Answer: Two parallel tangents can be drawn to an ellipse.

Q 2. What is the locus of the feet of perpendiculars drawn from foci upon a tangent of an ellipse?
Answer: The locus of the feet of perpendiculars on any tangent from the foci is the auxiliary circle of the ellipse.

Q 3.What is the area of the rectangle formed by the tangents parallel to the axes of the ellipse?
Answer: Length of the rectangle will be the same as the length of major axis which is and the breadth of the rectangle will be same as the length of the minor axis which is . Hence area of the rectangle formed by the tangents parallel to the axes of the ellipse will be

Q 4. Does the area of the rectangle formed by the tangents parallel to the axes of an ellipse change if the center of the ellipse is shifted to a new coordinate?
Answer: On shifting the center, all the lengths remain the same and hence the area will also remain the same.