Question

A straight line passes through a fixed point $P(h,k)$, then the locus of the foot of the perpendicular drawn to it from the origin is

• A. a circle
• B. an ellipse
• C. a parabola
• D. a hyperbola

Answer 1

The correct option is A a circle

Equation of a line (moving) passing through the point $P(h,k)$ is given by
$y-k=m(x-h)$ ...(i)
Let $A(\alpha ,\beta )$ be the foot of $\perp$er drawn from $O(0,0)$ to the line (i)
$\therefore$ Slope of $OA$ $\times$ Slope of line(i) $=-1$
$\Rightarrow \frac{\beta }{\alpha }\left(m\right)=-1$
$\therefore m=-\frac{\alpha }{\beta }$ ....(ii)
As $\alpha ,\beta$ lies on (i)
$\therefore \beta -k=m(\alpha -h)$ ......(iii)
Here '$m$' is variable so eliminating '$m$' from (ii), (iii) we get
$\beta -k=\frac{-\alpha }{\beta }(\alpha -h)$
$\Rightarrow {\beta }^2-k\beta =-{\alpha }^2+\alpha h\Rightarrow {\alpha }^2+{\beta }^2-\alpha h-k\beta =0$
$\therefore$ Locus of $A(\alpha ,\beta )$ is $x^2+y^2-xh-yk=0$
which represent a circle.
Hence (a) is correct answer.