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I need to find the tangent line to an ellipse by only knowing the slope of a parallel line to the tangent line

I need to find the tangent line to an ellipse by only knowing the slope of a parallel line to the tangent line

Grade:11

1 Answers

bharat bajaj IIT Delhi
askIITians Faculty 122 Points
10 years ago
Equation of ellipse : x²/a² + y²/b² = 1

Let the point of tangency, whose coordinates will also satisfy the equation of the tangent line, be (p,q).

The slope of the tangent line is equal to the value of the derivative at the point of tangency.

Derivating wrt to x,

2x/a² + 2yy'/b² = 0

y' = -(b²/a²)(x/y)

Therefore, evaluating y' when x = p and y = q,

m = -(b²/a²)(p/q)

Solving for, say, q, we have

q = -(b²/a²)(p/m)

Since (p, q) lies on the ellipse, it satisfies the ellipse's equation; thus,

p²/a² + q²/b² = 1

Substitute for q from the earlier result:

p²/a² + [-(b²/a²)(p/m)]²/b² = 1

This simplifies to

(p²/a²)[1 + (b²/(m²a²))] = 1
Thanks
Bharat Bajaj
IIT delhi
askiitians faculty

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