EllipseHardBloom L3
Question
The latus rectum of a conic section is the width of the function through the focus. The positive difference between the lengths of the latus rectum of $3y = x^{2} + 4x - 9$ and $x^{2} + 4y^{2} - 6x + 16y = 24$ is :
Options
A.$\frac{1}{2}$
B.2
C.$\frac{3}{2}$
D.$\frac{5}{2}$
Solution
Sol.(a) $(x + 2)^{2} = 3\left( y + \frac{13}{3} \right)$
⇒ Length of latus rectum of parabola, $L_{1} = 3$
$$\frac{(x - 3)^{2}}{49} + \frac{(y + 2)^{2}}{(49/4)} = 1$$
Lengths of latus rectum of ellipse,
$$\begin{matrix} L_{2} & \ = \frac{2b^{2}}{a} = \frac{2\left( \frac{49}{4} \right)}{7} = 7/2 \\ \left| L_{1} - L_{2} \right| & \ = \left| 3 - \frac{7}{2} \right| = 1/2 \end{matrix}$$
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