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Question

The locus of point of intersection of tangent to an ellipse at two points, sum of whose eccentric angle is constant is

A
Parabola
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B
Circle
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C
Ellipse
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D
Straight line
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Solution

The correct option is D Straight line
The equation of tangents at two points having eccentric angle θ1 and θ2 are

xacosθ1+ybsinθ1=1....(i)

and xacosθ2+ybsinθ2....(ii)

The point of intersection of Eqs. (i) and (ii) is

⎜ ⎜ ⎜ ⎜acos(θ1+θ22)cos(θ1θ22),bsin(θ1+θ22)cos(θ1θ22)⎟ ⎟ ⎟ ⎟

It is given that θ1+θ2=k=constant.

Therefore, if (x1,y1) is the point of intersection of Eqs. (i) and (ii), then

x1=acosk2cos(θ1θ22)

and

y1=bsink2cos(θ1θ22)

x1y1=abcot(k2)

y1=(bacot(k2))x1

(x1,y1) lies on the straight line y=(bacot(k2))x

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