Question

Three normals can be drawn from a point $(\sin \theta +2,\cos \theta )$; to the parabola $y^2=4ax-8a$ then values of $a$ lie in the interval

• A. $[-1,\frac{1}{2}]$
• B. $[-\frac{1}{2},0]$
• C. $[\frac{1}{2},1]$
• D. $[-\frac{1}{2},\frac{1}{2}]-\left\{0\right\}$

Answer 1

Answer: (D)

$[-\frac{1}{2},\frac{1}{2}]-\left\{0\right\}$

For $y^2=4a(x-2)$
If three normals can be drawn from the given point

(i) when $a>0$
$\sin \theta +2>2a+2$
$\Rightarrow a<\frac{1}{2}\sin \theta$
$\Rightarrow a\in [0,\frac{1}{2}]$
(ii) when, $a<0$
$\sin \theta +2<2a+2$
$\Rightarrow \sin \theta <2a$
$\Rightarrow a>\frac{1}{2}\sin \theta$
$\Rightarrow a\in [-\frac{1}{2},0]$
Hence, $a\in [-\frac{1}{2},\frac{1}{2}]-\left\{0\right\}$