Question

The locus point of intersection of tangents to the parabola $y^2=4ax$, the angle between them being always ${45}^{\circ }$
is

• A. $x^2-y^2+6ax-a^2=0$
• B. $x^2-y^2-6ax+a^2=0$
• C. $x^2-y^2+6ax+a^2=0$
• D. $x^2-y^2-6ax-a^2=0$

Answer 1

The correct option is C $x^2-y^2+6ax+a^2=0$

Equation of tangent is $y=mx+\frac{a}{m}$
$\Rightarrow m^{2}x-my+a=0\Rightarrow m_1+m_2=\frac{y}{x},m_1{m}_2=\frac{a}{x}$
$tan{45}^{\circ }=\left|\begin{array}{c}\frac{m_1-m_2}{1+m_1{m}_2}\end{array}\right|\Rightarrow {\left(\frac{y}{x}\right)}^2-4\left(\frac{a}{x}\right)={(1+\frac{a}{x})}^2$
$\Rightarrow x^2-y^2+6ax+a^2=0$