Given are the parabolas $y^{2} = 4 a (x - a)$ and $x^{2} = 4 a y$ ,find the equation of the circle with the line joining their foci as diameter.

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

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x^{2} - y^{2} - 2 a x - a y = 0

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x^{2} - y^{2} - 4 a x - 2 a y = 0

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

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Solution

The correct option is A $x^{2} + y^{2} - 2 a x - a y = 0$
$y^{2} = 4 A X,$ focus $S_{1}$ is $X = a, Y = 0$
or $x - a = a, y = 0$
$∴ S_{1}$ is $(2 a, 0)$ and $S_{2} = (0, a) .$
Circle on $S_{1} S_{2}$ as diameter is
$x (x - 2 a) + y (y - a) = 0$
or $x^{2} + y^{2} - 2 a x - a y = 0$

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Given are the parabolas y2=4a(x−a) and x2=4ay ,find the equation of the circle with the line joining their foci as diameter.

The correct option is A x2+y2−2ax−ay=0y2=4AX, focus S1 is X=a,Y=0or x−a=a,y=0∴S1 is (2a,0) and S2=(0,a).Circle on S1S2 as diameter isx(x−2a)+y(y−a)=0or x2+y2−2ax−ay=0

Given are the parabolas $y^{2} = 4 a (x - a)$ and $x^{2} = 4 a y$ ,find the equation of the circle with the line joining their foci as diameter.

The correct option is A $x^{2} + y^{2} - 2 a x - a y = 0$
$y^{2} = 4 A X,$ focus $S_{1}$ is $X = a, Y = 0$
or $x - a = a, y = 0$
$∴ S_{1}$ is $(2 a, 0)$ and $S_{2} = (0, a) .$
Circle on $S_{1} S_{2}$ as diameter is
$x (x - 2 a) + y (y - a) = 0$
or $x^{2} + y^{2} - 2 a x - a y = 0$