Let S and S? be the foci of the ellipse x 2 25 + y 2 9 = 1 and P be a point on the ellipse in the first quadrant. If ( S P ) 2 + ( S ' P ) 2 - S P · S ' P = 37 , then 2 + 2 is equal to: [2026]

Question

Let S and S′ be the foci of the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ and $P (α, β)$ be a point on the ellipse in the first quadrant. If ${(S P)}^{2} + {(S' P)}^{2} - S P \cdot S' P = 37,$ then $α^{2} + β^{2}$ is equal to: [2026]

Smart Achievers · Accepted Answer

(1)

$∵$ $P lies on ellipse \Rightarrow \frac{α^{2}}{25} + \frac{β^{2}}{9} = 1$

${(S P)}^{2} + {(S' P)}^{2} - S P \cdot S' P = 37,$ $P S + P S' = 2 a \Rightarrow P S + P S' = 10$

$∴$ ${(P S)}^{2} + {(P S')}^{2} - P S \cdot P S' = 37$

${(P S + P S')}^{2} - 3 P S \cdot P S' = 37$

$100 - 3 P S \cdot P S' = 37$

$3 P S \cdot P S' = 63 \Rightarrow P S \cdot P S' = 21$

$∵$ $P S & P S' are (5 \pm \frac{4}{5} α)$

$∴$ $P S \cdot P S' = 25 - \frac{16}{25} α^{2} = 21$

$\frac{16}{25} α^{2} = 4$

$α = \frac{5}{2} \Rightarrow α^{2} = \frac{25}{4}$

$∴$ $β^{2} = \frac{27}{4}$

$∴$ $α^{2} + β^{2} = \frac{52}{4} = 13$

Solution

Q.
Let S and S′ be the foci of the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ and $P (α, β)$ be a point on the ellipse in the first quadrant. If ${(S P)}^{2} + {(S' P)}^{2} - S P \cdot S' P = 37,$ then $α^{2} + β^{2}$ is equal to: [2026]

1 13

2 15

3 17

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