Question

Consider the parabola $X^2+4Y=0$. Let p = (a, b) be any fixed point inside the parabola and let 'S' be the focus of the parabola. Then the minimum value SQ + PQ as point Q moves on the parabola is

• A. $|1-a|$
• B. $\left|ab\right|+1$
• C. $\sqrt{a^2+b^2}$
• D. $1-b$

Answer 1

The correct option is D $1-b$

Let foot of perpendicular from Q to the directrix be N
$\Rightarrow$ SQ + PQ = QN + PQ is minimum it P, Q & N are collinear
So minimum value of SQ + PQ = PN = 1 - b
(PN is the distance of P from directrix. It will be equal to y coordinate minus 'a' if the parabola is $x^2=-4ay$ . Here, a = -1 and y coordinate = b)