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

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Solution

The correct option is C
Equation of tangent is $y = m x + \frac{a}{m}$
$\Rightarrow m^{2} x - m y + a = 0 \Rightarrow m_{1} + m_{2} = \frac{y}{x}, m_{1} m_{2} = \frac{a}{x}$
$t a n 45^{\circ} = ∣ ∣ \begin{matrix} \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} \end{matrix} ∣ ∣ \Rightarrow {(\frac{y}{x})}^{2} - 4 (\frac{a}{x}) = {(1 + \frac{a}{x})}^{2}$
$\Rightarrow x^{2} - y^{2} + 6 a x + a^{2} = 0$

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Similar questions

The locus of the point of intersection of perpendicular tangents to the circles $x^{2} + y^{2} = a^{2} a n d x^{2} + y^{2} = b^{2} i s$

x^{2} + y^{2} = a^{2}

y^{2} = 4 a x

θ_{1}, θ_{2}

x -

cos θ_{1} \cdot cos θ_{2} = k,

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, (a > b)

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