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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
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
|
is a tangent to ellipse 
and 
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:
|
(Standard vertical ellipse)
and 
are perpendicular to each other.
and 
, we get the following:
Example : Find the equation of tangent(s) to the ellipse 
whose inclination is 
. Also, find the point of tangency.
Solution :
|
(Standard horizontal ellipse)
is a tangent to ellipse 
is a tangent to ellipse 
, then 
, 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:
|
(Standard horizontal ellipse)
to be any point on the ellipse (other than the vertices).
, i.e., the midpoint of the segment of tangent 
between the axes.
by using parametric form, 
And 
and 
and 
by 
by 
is the correct answer.Example : Find the points on ellipse 
such that the tangent at each point makes equal angles with the axes.
Answer:
|
Step 1 : Let
Now, the slope of the tangent is as follows: ⇒ |
|
Step 2 : ⇒ Also, Therefore, the coordinates of the required points are |
, then the equation of tangent at 
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.
| Related Concept Links | |
| Terms related to an ellipse | Auxiliary Circle of an ellipse |
| Vertical Standard Ellipse | Definition of Ellipse |
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