Tangent at a point of an ellipse:
(i) Let equation of ellipse be 
.
Slope of tangent to ellipse at a point (x1, y1) = 
Therefore the equation of tangent at (x1, y1) is 
i.e. T= 0
(ii) Equation of tangent at point q that is (a cosθ, b sinθ) is obtained by putting
x1 = a cosθ, y1 = b sinθ; 
EQUATION OF THE TANGENT IN THE TERMS OF ITS SLOPE; USING THE CONCEPT OF COMPARISON:
The equations of tangent to ellipse having slope m are y = mx ± 
for all the finite values of m.
Moreover the line touches ellipse 
.
Illustration: From the point P 2 tangents are drawn one each to ellipse
, If the tangents are perpendicular to each other, then find the locus of point P.
Solution: The tangent at (a cosθ, b sinθ) on is
....(i)
The tangent at (a cosΦ, b sinΦ) on is
....(ii)
(i) and (ii) are perpendicular 
....(iii)
By eliminating f from (ii) and (iii)
Locus is (x2 + y2)2 = b2 (x + y)2 + a2(x - y)2
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