Question
A point on the parabola $${y^2} = 18x$$ at which the ordinate increases at twice the rate of the abscissa is
A.
$$\left( {\frac{9}{8},\frac{9}{2}} \right)$$
B.
$$\left( {2, - 4} \right)$$
C.
$$\left( {\frac{{ - 9}}{8},\frac{9}{2}} \right)$$
D.
$$\left( {2,4} \right)$$
Answer :
$$\left( {\frac{9}{8},\frac{9}{2}} \right)$$
Solution :
$$\eqalign{
& {y^2} = 18x \Rightarrow 2y\frac{{dy}}{{dx}} = 18 \Rightarrow \frac{{dy}}{{dx}} = \frac{9}{y} \cr
& {\text{Given }}\frac{{dy}}{{dx}} = 2 \Rightarrow \frac{9}{y} = 2 \Rightarrow y = \frac{9}{2} \cr
& {\text{Putting in }}{y^2} = 18x \Rightarrow x = \frac{9}{8} \cr
& \therefore {\text{Required point is }}\left( {\frac{9}{8},\frac{9}{2}} \right) \cr} $$