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.

Question

asked Jun 23, 2023 120k views

1 Answer

Tirolel · Answer 1 · 2023-06-26T14:49:41+0000

Answer:

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 ,

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

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

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

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

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