Which of the following statements is correct?
$I .$ Equation of the director circle with respect to the ellipse $16 x^{2} + 25 y^{2} = 400$ is $x^{2} + y^{2} = 41$
$I I .$ Equation of the tangent at $(3, 2)$ on the ellipse $x^{2} + 4 y^{2} = 25$ is $3 x + 8 y - 25 = 0$

Only I

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Only II

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Both I and II

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Neither I nor II

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Solution

The correct option is B Both I and II
I) Equation of director circle is $x^{2} + y^{2} = a^{2} + b^{2} = 25 + 16 = 41$
II) Equation of tangent to the ellipse $x^{2} + 4 y^{2} = 25$ is $S_{1} = 0$
$\Rightarrow h x + 4 k y = 25$
$\Rightarrow 3 x + 8 y - 25 = 0$

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Similar questions

4 x^{2} + 25 y^{2} = 100

x^{2} + y^{2} = 29

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Which of the following gives the equation of director circle of the ellipse $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ ?

4 x^{2} + 5 y^{2} = 20

List I	List II
A. Equation of auxiliary circle	1. $4 x + 5 y = 20$
B. Equation of director circle	2. $x^{2} + y^{2} = 5$
C. Equation of tangent at $(1, 1)$	3. $y - x + 9 = 0$
D. Equation of tangent with slope $1$	4. $x^{2} + y^{2} = 9$
	5. $5 x + 4 y = 20$
	6. $y = x + 3$

The equation of the directrices of the ellipse $16 x^{2} + 25 y^{2} = 400$ are

\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1

y = 3 x

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