Question

Let S and S' be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$. And points P and P' be the ends of focal chord through S. If SP=4, then S'P' is equal to

• A. $\frac{11}{3}$
• B. 21
• C. 7
• D. $\frac{22}{3}$

Answer 1

Answer: (D)

$\frac{22}{3}$

Answer 2

The equation of the ellipse is given by $\frac{x^2}{25}+\frac{y^2}{16}=1$.

This is in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a^2=25$ and $b^2=16$. So, $a=5$ and $b=4$.

Since $a > b$, the major axis is along the x-axis. The distance from the center to the foci is $c$, where $c^2 = a^2 - b^2 = 25 - 16 = 9$. Thus, $c=3$.

The foci are $S = (c, 0) = (3, 0)$ and $S' = (-c, 0) = (-3, 0)$. The eccentricity of the ellipse is $e = \frac{c}{a} = \frac{3}{5}$.

Let P and P' be the ends of a focal chord through S. Let S be the focus $(3,0)$. For any point on the ellipse, the sum of the distances from the two foci is constant and equal to $2a$. So, for point P on the ellipse, $SP + S'P = 2a$. We are given SP = 4 and $a=5$. $4 + S'P = 2(5) = 10$. $S'P = 10 - 4 = 6$.

Similarly, for point P' on the ellipse, $SP' + S'P' = 2a$. $SP' + S'P' = 10$. To find S'P', we need to find SP'.

P, S, and P' are collinear and S is the focus. P and P' are the ends of the focal chord through S. Let the points be P and P'. The distances from the focus S to these points are SP and SP'. There is a harmonic relationship between the segments of a focal chord. If a focal chord through S has endpoints P and P', then $\frac{1}{SP} + \frac{1}{SP'} = \frac{2}{l}$, where $l$ is the semi-latus rectum. The semi-latus rectum is $l = \frac{b^2}{a} = \frac{16}{5}$. So, $\frac{1}{SP} + \frac{1}{SP'} = \frac{2}{16/5} = \frac{10}{16} = \frac{5}{8}$. We are given SP = 4.

$\frac{1}{4} + \frac{1}{SP'} = \frac{5}{8}$. $\frac{1}{SP'} = \frac{5}{8} - \frac{1}{4} = \frac{5}{8} - \frac{2}{8} = \frac{3}{8}$. So, $SP' = \frac{8}{3}$.

Now we use the property $SP' + S'P' = 2a$ for the point P'. We have $SP' = 8/3$ and $2a = 10$. $\frac{8}{3} + S'P' = 10$. $S'P' = 10 - \frac{8}{3} = \frac{30 - 8}{3} = \frac{22}{3}$.

Therefore, the value of S'P' is $\frac{22}{3}$.