Let A(α, 0) and B(0, β) be the points on the line 5x + 7y = 50. ​ Let the point I divide the line segment

Question

Let \(A(\alpha, 0)\) and \(B(0, \beta)\) be the points on the line \(5 x+7 y=50\). Let the point \(P\) divide the line segment \(A B\) internally in the ratio \(7: 3\). Let \(3 x-25=0\) be a directrix of the ellipse \(E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and the corresponding focus be \(S\). If from \(S\), the perpendicular on the x-axis passes through \(\mathrm{P}\), then the length of the latus rectum of \(\mathrm{E}\) is equal to

(1) \(\frac{25}{3}\)

(2) \(\frac{32}{9}\)

(3) \(\frac{25}{9}\)

(4) \(\frac{32}{5}\)

Answer 1

Correct option is (4) \(\frac{32}{5}\)

\(\left.\begin{array} {l}A = (10, 0)\\B = \left(0, \frac{50}7\right)\end{array}\right\} P = (3, 5)\)

\(\mathrm{ae}=3\)

\(\frac{\mathrm{a}}{\mathrm{e}}=\frac{25}{3}\)

\(\mathrm{a}=5\)

\(\mathrm{b}=4\)

Length of \(L R=\frac{2 b^{2}}{a}=\frac{32}{5}\)