Points A(−1, y) and B(5, 7) lie on a circle with centre O(2, −3y).

Question:

Points A(−1, y) and B(5, 7) lie on a circle with centre O(2, −3y). Find the values of y.

Solution:

The given points are A(−1, y), B(5, 7) and O(2, −3y).
Here, AO and BO are the radii of the circle. So

$A O=B O \Rightarrow A O^{2}=B O^{2}$

$\Rightarrow(2+1)^{2}+(-3 y-y)^{2}=(2-5)^{2}+(-3 y-7)^{2}$

$\Rightarrow 9+(4 y)^{2}=(-3)^{2}+(3 y+7)^{2}$

$\Rightarrow 9+16 y^{2}=9+9 y^{2}+49+42 y$

$\Rightarrow 7 y^{2}-42 y-49=0$

$\Rightarrow y^{2}-6 y-7=0$

$\Rightarrow y^{2}-7 y+y-7=0$

$\Rightarrow y(y-7)+1(y-7)=0$

$\Rightarrow(y-7)(y+1)=0$

$\Rightarrow y=-1$ or $y=7$

Hence, y = 7 or y = −1.

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