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  • If P(h, k) be a point on the parabol which is nearest to the point Q(0,33), then the distance of P from

If P(h, k) be a point on the parabolx=4y^{2} which is nearest to the point Q(0,33), then the distance of P from the directrix of the

parabola y y^{2} = 4(x + y)  is equal to :

Option: 1

2


Option: 2

6


Option: 3

8


Option: 4

4


Answers (1)

best_answer

Equation of normal of the parabola x=4y^{2}

At a point \mathrm{P}\left(\frac{\mathrm{t}^2}{16}, \frac{2 \mathrm{t}}{16}\right) is

y+t x=\frac{2 t}{16}+\frac{1}{16} t^3

Normal pass through (Q=0,33) then

\begin{aligned} & 33=\frac{t}{8}+\frac{t^3}{16} \\ & \Rightarrow t^3+2 t-528=0 \\ & \Rightarrow(t-8)\left(t^2+8+166\right)=0 \\ & \Rightarrow t=8 \end{aligned}

Point P is (4, 1)

Given parabola is y^2=4(x+y)

                  \begin{aligned} & y^2-4 y=4 x \\ & (y-2)^2=4(x+1) \end{aligned}

   directrix is x + 1 = -1

                              x=-2

Distance ofP(4,1) from the directrix x = -2 is 6.

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Rishabh

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