    Question

# If the tangent drawn at point P(t2,2t) on the parabola y2=4x is same as the normal drawn at point Q(√5cosθ,2sinθ) on the ellipse 4x2+5y2=20, then

A
Q(1,45)
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B
Q(1,45)
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C
P(15,25)
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D
P(15,25)
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Solution

## The correct options are A Q≡(−1,4√5) B Q≡(−1,−4√5) C P≡(15,−2√5) D P≡(15,2√5) The equation of the tangent at P(t2,2t) on the parabola y2=4x is : ⇒x−ty+t2=0 …(1) Equation of the normal at point Q(√5cosθ,2sinθ) on the ellipse 4x2+5y2=20 is : (√5secθ)x−(2 cosec θ)y=5−4 ⇒(√5secθ)x−(2 cosec θ)y=1 …(2) Given that equations (1) and (2) represent the same line. ⇒√5secθ1=−2 cosec θ−t=−1t2 ⇒t=2√5cotθ and t=−12sinθ ⇒2√5cotθ=−12sinθ ⇒4cosθ=−√5sin2 θ ⇒4cosθ=−√5(1−cos2θ) ⇒√5cos2θ−4cosθ−√5=0 ⇒(cosθ−√5)(√5cosθ+1)=0 ⇒cosθ=−1√5 or cosθ=√5 (not possible) ⇒cosθ=−1√5⇒sinθ=±2√5 ⇒t=∓1√5 Hence, Q≡(−1,±4√5),P≡(15,∓2√5)  Suggest Corrections  0      Similar questions  Explore more