Question

Three normals to the parabola $y^2=x$ are drawn through a point $(C,0)$, then

• A. $C=\frac{1}{4}$
• B. $C=\frac{1}{2}$
• C. $C>\frac{1}{2}$
• D. $C<\frac{1}{2}$

Answer 1

Answer: (C)

$C>\frac{1}{2}$

Three normals to parabola $y^2=x$ are drawn through point $(C,0)$ then $C$ is

We know one normal is $x-$ axis

And for normal for parabola $y^2=4ax$ is

$y=mx-2am-a{m}^3$

Here $a=\frac{1}{4}$

$y=mx-\frac{m}{2}-\frac{m^3}{4}$

either $m=0$

$\frac{m^2}{4}+\frac{1}{2}-C=0(m\ne 0)m^2=4(C-\frac{1}{2})(m\ne 0)m^2\ge 0(m\ne 0)4(C-\frac{1}{2})\ge 0(C>\frac{1}{2})\therefore (m\ne 0)$