Question

The length of the chord of the parabola $y^2=12x$ passing through the vertex and making an angle of ${60}^0$ with axis of $x$ is:

• A. $8$
• B. $4$
• C. $\frac{16}{3}$
• D. $Noneofthese$

Answer 1

The correct option is A $8$

Solution:- (A) $8$

$y^2=4ax.....\left(1\right)$

Vertex $\equiv (0,0)$

Slope, $m=\tan \theta$

$\therefore$ Equation of the chord is-

$y=\left(\tan \theta \right)x$

Substituting the value of $y$ in ${eq}^n\left(1\right)$, we have

${\tan }^2\theta \left(x^2\right)=4ax$

$\Rightarrow x=\frac{4a}{\tan \theta }$

Again substituting the value of $x$ in ${eq}^n\left(1\right)$, we hve

$y=\frac{4a}{\tan \theta }$

$\therefore$ Length of the chord $\left(l\right)=\sqrt{{\left(\frac{4a}{{\tan }^2\theta }\right)}^2+{\left(\frac{4a}{\tan \theta }\right)}^2}$

$\Rightarrow l=\frac{4a}{{\tan }^2\theta }\sqrt{1+{\tan }^2\theta }$

$\Rightarrow l=\frac{4a}{{\tan }^2\theta }\sec \theta$

$\Rightarrow l=\frac{4a\cos \theta }{{\sin }^2\theta }$

Given that the equation of parabola is $y^2=12x$

$\therefore 4a=12$

Also given that $\theta =60°$

$\therefore l=\frac{12\cos 60°}{{\sin }^{2}60°}$

$\Rightarrow l=\frac{12\times \left(\frac{1}{2}\right)}{{\left(\frac{\sqrt{3}}{2}\right)}^2}$

$\Rightarrow l=8\text{units}$

Hence the correct answer is $8$.