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 5/e² is equal to ____

Let (h, k) lie on the circle x² + y² = 4 and (2h + 1, 3k + 2) lie on an ellipse with eccentricity e. Find the value of 5/e²

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

Correct Answer: 9

Explanation

Given \((h,k)\) lies on the circle

\[ h^2+k^2=4 \]

The transformed point is \((2h+1,3k+2)\). Subtracting the translation, let

\[ X=2h,\quad Y=3k \]

Then

\[ \frac{X^2}{4}+\frac{Y^2}{9}=h^2+k^2=4 \]

Dividing throughout by \(4\),

\[ \frac{X^2}{16}+\frac{Y^2}{36}=1 \]

This represents an ellipse with

\[ a^2=36,\quad b^2=16 \]

So,

\[ e^2=1-\frac{b^2}{a^2}=1-\frac{16}{36}=\frac{5}{9} \]

Hence,

\[ \frac{5}{e^2}=\frac{5}{5/9}=9 \]

Therefore, the correct answer is 9.

Explanation

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