Let A(, 0) and B(0, ) be the points on the line 5x + 7y = 50. Let the point P divide the line segment AB internally in the ratio 7 : 3. Let 3x – 25 = 0 be a directrix of the ellipse , and the corresponding focus be S. If from S, the perpendicular on the x

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Solution

Q.
Let A( $α$ , 0) and B(0, $β$ ) be the points on the line 5x + 7y = 50. Let the point P divide the line segment AB internally in the ratio 7 : 3. Let 3x – 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 P, then the length of the latus rectum of E is equal to, [2024]

1 $\frac{32}{5}$

2 $\frac{32}{9}$

3 $\frac{25}{9}$

4 $\frac{25}{3}$

Ans.
(1)

A( $α$ , 0) and B(0, $β$ ) be the points on the line 5x + 7y = 50

$\Rightarrow α = 10 and β = \frac{50}{7}$

Using section formula, we have P(3, 5).

Directrix : $x = \frac{25}{3} = \frac{a}{e}$

The equation of the line passing through P(3, 5) and perpendicular to x-axis is x = 3.

$∵$ The perpendicular is also passes through S.

$∴$ ae = 3

$a \times \frac{3}{25} a = 3 or a^{2} = 25 \Rightarrow a = 5$

$b^{2} = 25 (1 - \frac{9}{25}) = 16$

$∴$ Length of latus rectum $= \frac{2 b^{2}}{a} = \frac{2 \times 16}{5} = \frac{32}{5}$ .

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