Question

Find the length of focal chord AB of parabola when (i) length of latus rectum is 8, 'S' be the focus and $\frac{AS}{BS} = \frac{3}{4}$

• A. 49/6
• B. 25/3
• C. 36/5
• D. 16/3

Answer 1

Answer: (A)

49/6

Answer 2

Let the parabola be $y^2 = 4ax$. The length of the latus rectum is $4a = 8$, so $a = 2$. For a focal chord AB, let $AS = l_1$ and $BS = l_2$. We are given $\frac{l_1}{l_2} = \frac{3}{4}$. The relation for a focal chord is $\frac{1}{AS} + \frac{1}{BS} = \frac{1}{a}$. Substituting the ratio, let $l_1 = 3k$ and $l_2 = 4k$. $\frac{1}{3k} + \frac{1}{4k} = \frac{1}{2}$ $\frac{4+3}{12k} = \frac{1}{2}$ $\frac{7}{12k} = \frac{1}{2}$ $12k = 14 \implies k = \frac{14}{12} = \frac{7}{6}$ Then $AS = l_1 = 3 \times \frac{7}{6} = \frac{7}{2}$ and $BS = l_2 = 4 \times \frac{7}{6} = \frac{14}{3}$. The length of the focal chord AB is $AS + BS = \frac{7}{2} + \frac{14}{3} = \frac{21 + 28}{6} = \frac{49}{6}$.