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  • A circle passes through the point (3,4) and cuts the circle <img alt="\mathrm{x^2+y^2=a^{2}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5E2&plus;y%5E2%3Da%5E%7B2%7

A circle passes through the point (3,4) and cuts the circle \mathrm{x^2+y^2=a^{2}} orthogonally.The locus of its centre is a straight line.If the distance of the straight line from the origin is 817,then \mathrm{a^{2}} is equal to ________.

Option: 1

8145


Option: 2

3267


Option: 3

1425


Option: 4

8342


Answers (1)

best_answer

Let the equation of the circle be \mathrm{x^2+y^2+2 g x+2 f y+c=0},since Since it passes through (3, 4), 6g + 8f + c + 25 = 0. As it cuts the circle \mathrm{x^2+y^2=a^{2}} orthogonally.

\mathrm{ \therefore \quad 2 g \times 0+2 f \times 0=c-a^2 \Rightarrow c=a^2 \Rightarrow 6 g+8 f+a^2+25=0}

\mathrm{ Locus\; of\; the\; centre (-g,-f) is \; 6 x+8 y-\left(a^2+25\right)=0}

Distance of the line from the origin is

\mathrm{817=\frac{a^2+25}{\sqrt{36+64}} \Rightarrow a^2+25=8170 \Rightarrow a^2=8145}

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