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Question

The tangent and normal to the ellipse x2+4y2=4 at a point P(θ) on it meet the major axis in Q and R respectively. If QR=2, the eccentric angle θ of P is given by

A
cosθ=±23
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B
sinθ=±23
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C
tanθ=±23
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D
cotθ=±23
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Solution

The correct option is C cosθ=±23
The equation of tangent to ellipse is bxcosθ+aysinθ=ab
It meets major axis at x=acosθ
The equation of normal to ellipse is axsecθbycosecθ=a2b2
It meets the major axis at x=a2b2asecθ
Here a=2 and b=1
So, point Q is (2cosθ,0) and point R is (32secθ,0)
The distance between them QR=3cosθ22cosθ=2
3cos2θ4cosθ4=0
cosθ=23

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