A circle, ${{C}_{1}}$, passes through the points $A\left( 4,0 \right)$ and \(B\left( 0,8 \right)

Question

A circle, ${{C}_{1}}$, passes through the points $A\left( 4,0 \right)$ and $B\left( 0,8 \right)$. Its centre, $P$, lies on the line $y=x+1$. Show that the centre of the circle, ${{C}_{1}}$ is $\left( 4,5 \right)$. Hence find the equation of the circle, ${{C}_{1}}$. A second circle, ${{C}_{2}}$, with the equation ${{x}^{2}}+{{y}^{2}}+ax+by+32=0$, has the same centre, $P$, as circle ${{C}_{1}}$. Find the value of $a$ and of $b$. Show that the circle ${{C}_{2}}$ lies within the circle ${{C}_{1}}$.

Accepted Answer

(a) Centre $P=(4,5)$, $C_1:(x-4)^2+(y-5)^2=25$ (b) $a=-8$, $b=-10$ (c) $r_2=3<5=r_1$, so $C_2$ lies within $C_1$

Coordinate Geometry - Finding the Centre and Radius of a Circle

Coordinate Geometry - Finding the Centre and Radius of a Circle

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