Parabola Question 4
Question 4 - 30 January - Shift 1
If $P(h, k)$ be point on the parabola $x=4 y^{2}$, which is nearest to the point $Q(0,33)$, then the distance of $P$ from the directrix of the parabola $y^{2}=4(x+y)$ is equal to :
(1) 2
(2) 4
(3) 8
(4) 6
Show Answer
Answer: (4)
Solution:
Formula: Equation of normal at any Point in Parametric form, Equation of Directrix
Equation of normal
$y=-t x+2 a t+a t^{3}$
$y=-t x+\frac{2}{16} t+\frac{1}{16} t^{3}$
It passes through $(0,33)$
$33=\frac{t}{8}+\frac{t^{3}}{16}$
$t^{3}+2 t-528=0$
$(t-8)(t^{2}+8 t+66)=0$
$t=8$
$P(at^{2}, 2 at)=(\frac{1}{16} \times 64,2 \times \frac{1}{16} \times 8)=(4,1)$
Parabola :
$y^{2}=4(x+y)$
$\Rightarrow y^{2}-4 y=4 x$
$\Rightarrow(y-2)^{2}=4(x+1)$
Equation of directix :-
$x+1=-1$
$x=-2$
Distance of point $=6$
Ans. : (4)
