What is the angle subtended by the latus rect | vedclass

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(D) The equation of the parabola is $y^2 = ax$. Comparing this with the standard form $y^2 = 4Ax$,we get $4A = a$,so $A = \frac{a}{4}$.The coordinates

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None of these

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(D) The equation of the parabola is $y^2 = ax$. Comparing this with the standard form $y^2 = 4Ax$,we get $4A = a$,so $A = \frac{a}{4}$.The coordinates of the vertex are $(0, 0)$.The latus rectum is the chord passing through the focus $(A, 0)$ perpendicular to the axis of the parabola.The endpoints of the latus rectum are $(A, 2A)$ and $(A, -2A)$,which are $(\frac{a}{4}, \frac{a}{2})$ and $(\frac{a}{4}, -\frac{a}{2})$.Let the vertex be $O(0, 0)$,and the endpoints of the latus rectum be $L(\frac{a}{4}, \frac{a}{2})$ and $L'(\frac{a}{4}, -\frac{a}{2})$.The slope of $OL$ is $m_1 = \frac{a/2}{a/4} = 2$.The slope of $OL'$ is $m_2 = \frac{-a/2}{a/4} = -2$.Let $	heta$ be the angle between $OL$ and $OL'$. The angle $\alpha$ that $OL$ makes with the $x$-axis is $	an \alpha = 2$,so $\alpha = 	an^{-1}(2)$.The angle $	heta$ subtended at the vertex is $2\alpha = 2 	an^{-1}(2)$.Since $2 	an^{-1}(2) = \pi - 	an^{-1}(\frac{4}{3}) \approx 126.87^\circ$,which is not equal to $\frac{\pi}{2}$,$\frac{\pi}{3}$,or $\frac{\pi}{4}$.Thus,the correct option is $D$.

What is the angle subtended by the latus rectum of the parabola $y^2 = ax$ at its vertex?

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