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

Answer:

r = 25

c = 20

k = 9 or k = -5

Step-by-step 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 ,

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

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

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

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

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