Which of the following points lies on the loc | vedclass

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(A) The locus of the foot of the perpendicular drawn from the foci to any tangent of an ellipse is its auxiliary circle.For the ellipse $\frac{x^{2}}{

Vedclass · Accepted Answer

$(-1, \sqrt{3})$

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(A) The locus of the foot of the perpendicular drawn from the foci to any tangent of an ellipse is its auxiliary circle.For the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$,the auxiliary circle is $x^{2}+y^{2}=a^{2}$.Given the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$,we have $a^{2}=4$.Thus,the locus is $x^{2}+y^{2}=4$.We check the given points to see which satisfies $x^{2}+y^{2}=4$:$A: (-1)^{2}+(\sqrt{3})^{2} = 1+3=4$ (Satisfies)$B: (-1)^{2}+(\sqrt{2})^{2} = 1+2=3 
eq 4$$C: (-2)^{2}+(\sqrt{3})^{2} = 4+3=7 
eq 4$$D: (1)^{2}+(2)^{2} = 1+4=5 
eq 4$Therefore,the point $(-1, \sqrt{3})$ lies on the locus.

Which of the following points lies on the locus of the foot of the perpendicular drawn from any of the foci of the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ to any of its tangents?

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