Q. The set of points on the axis of the parabola $y^{2} - 2y - 4x + 5 = 0$ from which all the three normals to the parabola are real, is
A) $\{(x,1):x \geq 3\}$
B) $\{(x, - 1):x \geq 1\}$
C) $\{(x,3):x \geq 1\}$
D) $\{(x, - 3):x \geq 3\}$
Explanation
(a)
Given equation of parabola can be rewritten as
${(y - 1)}^{2} = 4(x - 1)$
Axis of parabola is $y = 1$, equation of normal is
$(y - 1) = m(x - 1) - 2m - m^{3}$
Let $(h,\ 1)$ is a point on it’s axis, then
$0 = m(h - 1) - 2m - m^{3}$
$\Rightarrow m^{2} = h - 3$
$\Rightarrow h \geq 3$ for real values of $m$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.