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Mathematics Question on Conic sections

Let PQPQ be a double ordinate of the parabola, y2=−4xy^2= - 4x, where P lies in the second quadrant. If R divides PQPQ in the ratio 2:12 : 1, then the locus of R is :

A

9y2=4x9y^2 = 4x

B

9y2=−4x9y^2 = - 4x

C

3y2=2x3y^2 = 2x

D

3y2=−2x3y^2 = - 2x

Answer

9y2=−4x9y^2 = - 4x

Explanation

Solution

LetP(−at12,2at1),Let P\left(-at^{2}_{1}, 2at_{1}\right),
Q(−at12,−2at1)Q\left(-at^{2}_{1}, -2at_{1}\right) and R(h,k)R\left(h, k\right)
By using section formula, we have
h=−at12,k=−2at13h=-at^{2}_{1}, k=\frac{-2at_{1}}{3}
k=−2at13k=-\frac{2at_{1}}{3}
⇒3k=−2at1\Rightarrow 3k=-2at_{1}
⇒9k2=4a2 t12=4a(−h)\Rightarrow 9k^{2}=4a^{2}\,t^{2}_{1}=4a\left(-h\right)
⇒9k2=−4ah\Rightarrow 9k^{2}=-4ah
⇒9k2=−4h\Rightarrow 9k^{2}=-4h
⇒9y2=−4x\Rightarrow 9y^{2}=-4x