Equation of Tangent to Ellipse

An ellipse is a curve in the plane which surrounds the two focal points such that the distances to the focal point remain constant for each point on the curve. A circle is said to be a special type of ellipse having both focal points at the same point. A line which intersects the ellipse at a point is called a tangent to the ellipse. The different forms of the tangent equation are given below:

• Slope form of a tangent to an ellipse

If the line y = mx + c touches the ellipse x² / a² + y² / b² = 1, then c² = a²m² + b². The straight line y = mx ∓ √[a²m² + b²] represents the tangents to the ellipse.

• Point form of a tangent to an ellipse

The equation of the tangent to an ellipse x² / a² + y² / b² = 1 at the point (x₁, y₁) is xx₁ / a² + yy₁ / b² = 1.

• The parametric form of a tangent to an ellipse

The equation of the tangent at any point (a cosɸ, b sinɸ) is [x / a] cosɸ + [y / b] sinɸ.

• Point of contact of the tangent to an ellipse

Line y = mx ∓ √[a²m² + b²] touches the ellipse x² / a² + y² / b² = 1 at (∓a²m / √[a²m² + b²]) , (∓b² / √[a²m² + b²]).

Equation of Tangent to Ellipse Problems

Example 1: What is the locus of the point of intersection of perpendicular tangents to the ellipse x² / a² + y² / b² = 1?

Solution:

Let the point be (h, k). The pair of tangents will be

(x² / a² + y² / b² − 1) (h² / a² + k² / b² − 1) = (hx / a² +yk / b² − 1)²

The pair of tangents will be perpendicular if coefficient of x² + coefficient of y² = 0

k² / a²b² + h² / a²b² = 1 / a² + 1 / b²

h² + k² = a² + b²

Replace (h, k) by (x, y)

x² + y² = a² + b².

Example 2: What is the condition for the line lx + my − n = 0 to be tangent to the ellipse x² / a² + y² / b² = 1?

Solution:

y = [−l / m] x + n / m is tangent to x² / a² + y² / b² = 1, if

n / m = ± √b² + [a² / (l / m)²] or

On simplification,

n² = m²b² + l²a².

Example 3: If any tangent to the ellipse x² / a² + y² / b² = 1 cuts off intercepts of length h and k on the axes, then a² / h² + b² / k² = ___________.

Solution:

The tangent at (a cosθ, b sinθ) to the ellipse is [(a cos θ) * x] / [a2] +[b sin θ] * y / b2 = 1

or x / (a / cos θ) + y / (b / sin θ) = 1

∴Intercepts are, h = a / cos θ, k = b sin θ

a² / h² + b² / k² = 1

Example 4: The equation of the tangents drawn at the ends of the major axis of the ellipse 9x² + 5y² − 30y = 0, are ___________.

Solution:

Change the equation 9x² + 5y² − 30y = 0 in standard form 9x² + 5 (y² − 6y) = 0

9x² + 5 (y² − 6y + 9) = 45

x² / 5 + (y − 3)² / 9 = 1

∵ a² < b², so the axis of the ellipse is on the y-axis.

At the y-axis, put x = 0, so we can obtain the vertex.

Then, 0 + 5y² − 30y = 0

y = 0, y = 6

Therefore, tangents of vertex y = 0, y = 6.

Example 5: The equation of tangent and normal at point (3, 2) of ellipse 4x² + 9y² = 36 are _________.

Solution:

Given, the equation of an ellipse is 4x² + 9y² = 36

The tangent at point (3, 2) is (3) * x / 9 + (−2) * y / 4 = 1 or x / 3 − y / 2 = 1

∴ Normal is x / 2 + y / 3 = k, and it passes through the point (3,2)

∴ 3 / 2 − 2 / 3 = k ⇒ k = 5 / 6

∴ Normal is, x / 2 + y / 3 = 5 / 6.

Example 6: Minimum area of the triangle by any tangent to the ellipse x² / a² + y² / b² = 1 with the coordinate axes is _____________.

Solution:

The equation of tangent at (a cos θ, b sin θ) is [x / a] cos θ + [y / b] sin θ = 1

P = (a / cos θ, 0)

Q = (0, b / sin θ)

Area of OPQ = 1 / 2 ∣(a / cos θ) (b / sin θ)∣ = ab / |sin 2θ|

(Area)_min = ab

Example 7: The eccentric angles of the extremities of latus recta of the ellipse

Solution: 

Coordinates of any point on the ellipse

whose eccentric angle is θ are (a cos θ, b sin θ). 

Because 

Thus, the answer is option C.

Example 8: The length of the axes of the conic

Solution:

 Given that, the equation of the conic

Here

The length of the axes of the conic are 1, 2/3.

Thus, the answer is option C.

Example 9: The coordinates of the foci of the ellipse

A) (1, 2), (3, 4)                               

B) (1, 4), (3, 1)

C) (1, 1), (3, 1)                               

D) (2, 3), (5, 4)

Solution:

Foci are

Thus, the answer is option C.

Definitions of an Ellipse

Visualising Ellipses