Question

The points P (22, 15), Q (−13, c) and R (k, 24) all lie on a circle, centre (2, 0). Find the radius of the circle and the possible values of the constants c and k.

Answer 1

r = 25

c = 20

k = 9 or k = -5

Explanation:

Circle of center A(2, 0) and radius r.

P (22, 15), Q(−13, c) and R (k, 24) all lie on a circle,

`r=AP=\sqrt{\left( 22-2\right)^(2) +\left( 15-0\right)^(2) }`

`=25`

Q(−13, c) lies on the circle then ,

`AQ=r\Longrightarrow AQ^(2)=r^(2)\Longrightarrow (-13-2)^(2)+(c-0)^2=25^2=625`

`\Longrightarrow 225+c^(2)=625`

`\Longrightarrow c=20`

R (k, 24) lies on the circle then ,

`AR=r\Longrightarrow AR^(2)=r^(2)\Longrightarrow (k-2)^(2)+(24-0)^2=25^2=625`

`\Longrightarrow (k-2)^(2)+576=625`

`\Longrightarrow (k-2)^(2)=49`

`\Longrightarrow k-2=7\ \ or \ \ k-2=-7`

`\Longrightarrow k=9\ \ or \ \ k=-5`