If Y-axis is the directrix of the ellipse with eccentricity $e = 1 / 2$ and corresponding focus is at $(3, 0)$ , then the equation to its auxiliary circle is

x^{2} + y^{2} - 8 x + 12 = 0

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x^{2} + y^{2} - 8 x - 12 = 0

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x^{2} + y^{2} - 8 x + 9 = 0

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

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Solution

The correct option is A $x^{2} + y^{2} - 8 x + 12 = 0$
Let center of the ellipse be $(h, k)$

Given that y-axis is the directrix.

$\Rightarrow x = 0$

We know that directrix is $x = h - \frac{a}{e}$

So, $\Rightarrow h - \frac{a}{e} = 0$

$\Rightarrow h = \frac{a}{e}$

$\Rightarrow h = \frac{a}{1 / 2}$

$\Rightarrow h = 2 a$

We know that focus $= (h - a e, k)$

$\Rightarrow (h - a e, k) = (3, 0)$

$\Rightarrow h - a e = 3$

$\Rightarrow 2 a - (\frac{a}{2}) = 3$

$\Rightarrow \frac{3 a}{2} = 3$

$\Rightarrow a = 2$

$\Rightarrow a^{2} = 4$

We know that $b^{2} = a^{2} (1 - e^{2})$

$\Rightarrow h = 2 a = 4; k = 0$

$\Rightarrow b^{2} = 4 (1 - \frac{1}{4})$

$\Rightarrow b^{2} = 3$

Thus equation of ellipse $\Rightarrow \frac{(x - 4)^{2}}{4} + \frac{y^{2}}{3} = 1$

Hence, equation of auxiliary circle is $\Rightarrow (x - h)^{2} + (y - k)^{2} = a^{2}$

$\Rightarrow (x - 4)^{2} + y^{2} = 4$

$\Rightarrow x^{2} + y^{2} - 8 x + 16 = 4$

$\Rightarrow x^{2} + y^{2} - 8 x + 12 = 0$

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Similar questions

Q. The equation of the parabola with focus (0, 0) and directrix x + y = 4 is
(a) x² + y² − 2xy + 8x + 8y − 16 = 0
(b) x² + y² − 2xy + 8x + 8y = 0
(c) x² + y² + 8x + 8y − 16 = 0
(d) x² − y² + 8x + 8y − 16 = 0

The circle $x^{2} + y^{2} - 8 x = 0$ and hyperbola $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$ intersect at the points A and B.
Equation of the circle with AB as its diameter is

Q. The circle

x^{2} + y^{2} - 8 x = 0

and hyperbola

\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1

intersect at point

A

and

B

. The equation of circle with

A B

as diameter is:

Q. The equation of the circle concentric with the circle

x^{2} + y^{2} - 8 x + 6 y - 5 = 0

and passing through the point

(- 2, - 7)

Circles	Number of common tangents
I: $x^{2} + y^{2} = 4$ $x^{2} + y^{2} - 8 x + 12 = 0$	a) $0$
II: $x^{2} + y^{2} = 1$ $x^{2} + y^{2} - 2 x - 6 y + 6 = 0$	b) $1$
III: $x^{2} + y^{2} = 16$ $x^{2} + y^{2} - 8 x + 6 y - 56 = 0$	c) $4$
IV: $x^{2} + y^{2} - 2 x - 6 y + 9 = 0$ $x^{2} + y^{2} + 6 x - 2 y + 1 = 0$	d) $3$

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If Y-axis is the directrix of the ellipse with eccentricity e=1/2 and corresponding focus is at (3,0), then the equation to its auxiliary circle is

If Y-axis is the directrix of the ellipse with eccentricity $e = 1 / 2$ and corresponding focus is at $(3, 0)$ , then the equation to its auxiliary circle is