Let ( h , k ) lie on the circle C : x 2 + y 2 = 4 and the point ( 2 h + 1 , ? 3 k + 2 ) lie on an ellipse with eccentricity e . Then the value of 5 e 2 is equal to _________ [2026]

Question

Let $(h, k)$ lie on the circle $C : x^{2} + y^{2} = 4$ and the point $(2 h + 1, 3 k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\frac{5}{e^{2}}$ is equal to _________ [2026]

Smart Achievers · Accepted Answer

(9)

$Let P \equiv (2 \cos θ, 2 \sin θ)$

$∴$ $coordinates of Q = (4 \cos θ + 1, 6 \sin θ + 3)$

$∴$ $locus of Q is {(\frac{x - 1}{4})}^{2} + {(\frac{y - 3}{6})}^{2} = 1$

$∴$ $e^{2} = 1 - \frac{16}{36} = \frac{5}{9}$

$∴$ $\frac{5}{e^{2}} = 9$

Solution

Q.
Let $(h, k)$ lie on the circle $C : x^{2} + y^{2} = 4$ and the point $(2 h + 1, 3 k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\frac{5}{e^{2}}$ is equal to _________ [2026]

Ans.
(9)

$Let P \equiv (2 \cos θ, 2 \sin θ)$

$∴$ $coordinates of Q = (4 \cos θ + 1, 6 \sin θ + 3)$

$∴$ $locus of Q is {(\frac{x - 1}{4})}^{2} + {(\frac{y - 3}{6})}^{2} = 1$

$∴$ $e^{2} = 1 - \frac{16}{36} = \frac{5}{9}$

$∴$ $\frac{5}{e^{2}} = 9$

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Solution

Q. Let (h,k) lie on the circle C:x2+y2=4 and the point (2h+1, 3k+2) lie on an ellipse with eccentricity e. Then the value of 5e2 is equal to _________ [2026]

Ans. (9) Let P≡(2cosθ, 2sinθ) ∴ coordinates of Q=(4cosθ+1, 6sinθ+3) ∴ locus of Q is (x-14)2+(y-36)2=1 ∴ e2=1-1636=59 ∴ 5e2=9

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Q.
Let $(h, k)$ lie on the circle $C : x^{2} + y^{2} = 4$ and the point $(2 h + 1, 3 k + 2)$ lie on an ellipse with eccentricity $e$ . Then the value of $\frac{5}{e^{2}}$ is equal to _________ [2026]

Ans.
(9)

$Let P \equiv (2 \cos θ, 2 \sin θ)$

$∴$ $coordinates of Q = (4 \cos θ + 1, 6 \sin θ + 3)$

$∴$ $locus of Q is {(\frac{x - 1}{4})}^{2} + {(\frac{y - 3}{6})}^{2} = 1$

$∴$ $e^{2} = 1 - \frac{16}{36} = \frac{5}{9}$

$∴$ $\frac{5}{e^{2}} = 9$