Question

From a point $(\sin \theta ,\cos \theta )$ if three normals can be drawn to the parabola $y^2=4ax$, then the value of $a$ is

• A. $(\frac{1}{2},1)$
• B. $[-\frac{1}{2},0)$
• C. $[\frac{1}{2},1]$
• D. $(-\frac{1}{2},0)\cup (0,\frac{1}{2})$

Answer 1

Answer: (D)

$(-\frac{1}{2},0)\cup (0,\frac{1}{2})$

Point $(\sin \theta ,\cos \theta )$ lies on the circle $x^2+y^2=1$ for $\forall \theta \in R$.
Now three normals can be drawn to the parabola $y^2=4ax$ if $x>|2a|$
Hence, we must have $\cos \theta >|2a|$
$\Rightarrow 0<|2a|<1$
$\Rightarrow 0<|a|<\frac{1}{2}$

$\Rightarrow \frac{-1}{2}<a<0$ and $0<a<\frac{1}{2}$

$\Rightarrow a\in (-\frac{1}{2},0)\cup (0,\frac{1}{2})$