The locus of the middle point of the chords of the circle \( \Large x^{2}+y^{2}=a^{2} \) such that the chords pass through a given point \( \Large \left(x_{1},\ y_{1}\right) \) is:

A) \( \Large x^{2}+y^{2}-xx_{1}-yy_{1}=0 \)

B) \( \Large x^{2}+y^{2}=x^{2}_{1}+y^{2}_{1} \)

C) \( \Large x+y=x_{1}+y'_{2} \)

D) \( \Large x+y=x^{2}_{1}+y^{2}_{1} \)

Correct answer:
A) \( \Large x^{2}+y^{2}-xx_{1}-yy_{1}=0 \)

Description for Correct answer:

Let \( \Large P \left(x_{1}, y_{1}\right) \) be the point, then the chord of contact of tangents drawn from P to the circle

\( \Large x^{2}+y^{2}=a^{2}\ is\ xx_{1}+yy_{1}=a^{2} \)

\( \Large \therefore\ x^{2}+y^{2} = a^{2} \left(\frac{xx_{1}+yy_{1}}{a^{2}}\right) \)

=> \( \Large x^{2}+y^{2}-xx_{1}-yy_{1}=0 \)



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