The coordinate of the point on the parabola y2=8 x which is at minimum distance from the circle x 2+( y +6)2=1 are

Question

The coordinate of the point on the parabola y2=8 x which is at minimum distance from the circle x 2+( y +6)2=1 are (A) (2,-4) (B) (18,-12) (C) (2,4) (D) (-2,-4)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/ba2baecd566afec05e249810cd9f90a5-.png" /> Let P (2 t 2, 4 t ) be any point on the parabola, equation of normal at P (2 t 2, 4 t ) y+t x=4 t+2 t3 this normal passes through (0,-6) -6+0=4 t+2 t3 t3+2 t+3=0 (t+1)(t2-t+3)=0 t=-1, t2-t+3=0 D < 0 P(2,-4)

Q. The coordinate of the point on the parabola $y^{2} = 8 x$ which is at minimum distance from the circle $x^{2} + (y + 6)^{2} = 1$ are

$(2, - 4)$

$(18, - 12)$

$(2, 4)$

$(- 2, - 4)$

Let $P (2 t^{2}, 4 t)$ be any point on the parabola, equation of normal at $P (2 t^{2}, 4 t)$ $y + t x = 4 t + 2 t^{3}$
this normal passes through $(0, - 6)$
$- 6 + 0 = 4 t + 2 t^{3}$
$t^{3} + 2 t + 3 = 0$
$(t + 1) (t^{2} - t + 3) = 0$
$t = - 1, t^{2} - t + 3 = 0$
$D < 0$
$P (2, - 4)$

Q. The coordinate of the point on the parabola y2=8x which is at minimum distance from the circle x2+(y+6)2=1 are

(2,−4)

(18,−12)

(2,4)

(−2,−4)