Question

Tangents are drawn from any point on the line $x+4a=0$ to the parabola $y^2=4ax$. Then the angle subtended by the chord of contact at the vertex will be .

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is A $\frac{\pi }{2}$

Equation of a tangent of the parabola $y^2=4\mathrm{ax}$ having slope m is $y=\mathrm{mx}+\frac{a}{m}$.
Let there be a point P($-4a,y_1$) on the line x+4a=0.
If the tangent $y=\mathrm{mx}+\frac{a}{m}$ passes through ($-4a,y_1$), then $y_1=-4am+\frac{a}{m}\to 4a{m}^2+m{y}_1-a=0$
Let the roots of these equation be $m_1,m_2$.
Differentiating the given equation of parabola, we get $2yy\text{'}=4a\leftrightarrow y=\frac{2a}{y\text{'}}$
y-coordinates of points say A, B from where these tangents are drawn are $\frac{2a}{m_1},\frac{2a}{m_2}$
We have to find the angle between OA and OB where O is the origin.
$y^2=4ax\to \frac{y}{x}=\frac{4a}{y}$
Slope of a line passing through origin and a point on the parabola($x_1,y_1$) is $\frac{4a}{y_1}$.
Angle between the lines of slopes $m_1,m_2$ is ${\tan }^{-1}\left(\right(\frac{m_1-m_2}{1+m_1{m}_2}\left|\right)$.
Here, slopes are $\frac{4a}{\frac{2a}{m_1}}=2m_1,\frac{4a}{\frac{2a}{m_2}}=2m_2$.
So, the angle subtended is ${\tan }^{-1}\left(\right(\frac{2m_1-2m_2}{1+4m_1{m}_2}\left|\right)$.
$m_1{m}_2=\frac{-1}{4}\left(\mathrm{Product}\mathrm{of}\mathrm{the}\mathrm{roots}\right)$
Denominator becomes zero.
So, the angle subtended is ${90}^0$.