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 parabola. Then the minimum value of $SQ+PQ$ as point $Q$ moves on parabola is:

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

Answer 1

The correct option is D $1-b$

Let the foot of perpendicular from $Q$ to the directrix be $N$.
Now, since focal distance of $Q=QN$
So, for $SQ+PQ$ to be minimum $QN+PQ$ should also be minimum since $SQ=QN$
Here, $S,P$ are fixed points and $Q,N$ are variable points.
$\therefore$ For $QN+PQ$ to be minimum, $P,Q,N$ should be collinear. So minimum value of $SQ+PQ=PQ+QN=PN=1-b$