If the circle C_1 : x^{2} + y^{2} = 16 inters | vedclass

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Question

(C) Let the center of circle $C_2$ be $(h, k)$. Its equation is $(x - h)^{2} + (y - k)^{2} = 5^{2} = 25$.The equation of $C_1$ is $x^{2} + y^{2} = 16$

Vedclass · Accepted Answer

$(\pm 9/5, \pm 12/5)$

Vedclass · Answer

(C) Let the center of circle $C_2$ be $(h, k)$. Its equation is $(x - h)^{2} + (y - k)^{2} = 5^{2} = 25$.The equation of $C_1$ is $x^{2} + y^{2} = 16$.The equation of the common chord is given by $S_1 - S_2 = 0$,which is $2hx + 2ky = h^{2} + k^{2} - 9$.The length of the common chord is maximum when the chord passes through the center of $C_1$,i.e.,$(0, 0)$.Thus,$h^{2} + k^{2} - 9 = 0$,or $h^{2} + k^{2} = 9$.The slope of the common chord $2hx + 2ky = 9$ is $-h/k = 3/4$,which implies $k = -4h/3$.Substituting $k$ into $h^{2} + k^{2} = 9$,we get $h^{2} + (-4h/3)^{2} = 9$,so $h^{2} + 16h^{2}/9 = 9$,which gives $25h^{2}/9 = 9$,so $h^{2} = 81/25$.Thus,$h = \pm 9/5$. If $h = 9/5$,then $k = -4(9/5)/3 = -12/5$. If $h = -9/5$,then $k = 12/5$.Therefore,the possible centers are $(9/5, -12/5)$ and $(-9/5, 12/5)$.

If the circle $C_1 : x^{2} + y^{2} = 16$ intersects another circle $C_2$ of radius $5$ such that the length of the common chord is maximum and its slope is $3/4$,then the coordinates of the center of $C_2$ are:

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