If Y-axis is the directrix of the eillipse with eccentricty $e = 1 / 2$ 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 $x^{2} + y^{2} - 8 x + 12 = 0$
Distance between the focus and directrix is $\frac{a}{e} - a e = 3 ⟹ a = 2$ ; the center will be $\frac{a}{e}$ from directrix that is $(4, 0)$ .Auxilary circle is $(x - 4)^{2} + y^{2} = 4$

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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

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:

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 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 eillipse with eccentricty e=1/2 and corresponding focus is at (3,0), then the equation to its auxiliary circle is

The correct option is A x2+y2−8x+12=0Distance between the focus and directrix is ae−ae=3⟹a=2; the center will be ae from directrix that is (4,0).Auxilary circle is (x−4)2+y2=4

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

The correct option is A $x^{2} + y^{2} - 8 x + 12 = 0$
Distance between the focus and directrix is $\frac{a}{e} - a e = 3 ⟹ a = 2$ ; the center will be $\frac{a}{e}$ from directrix that is $(4, 0)$ .Auxilary circle is $(x - 4)^{2} + y^{2} = 4$