Math Assignment Help With Equations Of Tangent And Normal To An Ellipse

7.6.8 Equations of tangent and normal to an ellipse:

Theorem: The equation of tangent to the ellipse

x² + y² ‗ 1

a² b²

at a point (x1, y1) is

xx1 + yy1 ‗ 1

a² b²

and the equation of normal to the ellipse is x² + y² ‗ 1; at point (x1, y1) is

a² b²

(y – y1) ‗ (x – x1)

(y1/b²) (x1/ a²)

Proof: Let P(x1, y1) is a given point on the ellipse x<up>² + y² ‗  1 and Q(x2, y2) be a point on it.

a² b²

then the equation of the chord PQ is

y – y1 ‗ y2 – y1

x – x1 x2 – x1 …..(i)

as point p(x1, y1) and (x2, y2) lie on the given ellipse, therefore;

x1² + y1² ‗ 1 and x2² + y2²

a2 b2 a2 b2

On subtraction we get

x2² – x1² + y2² – y1² ‗ 1

a² b²

10x – 9y + 77 = 0

The equation of the normal at (-5, 3)

(77/2) [ x – (-5)] /-5 = (77/3) (y – 3)/3

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