Question

$AB$ is a diameter of a circle with centre $C(–2,5)$. If point $A$ is $(3,–7)$. Find:
(i) the length of radius $AC$.
(ii) the coordinates of $B$.

Answer 1

(i) Length of radius $AC=d(A,C)$

Co-ordinates of $A=(3,-7)$

Co-ordinates of $C=(-2,5)$

Here $x_1=3,y_1=-7,x_2=-2,y_2=5$

By distance formula, $d(A,C)=\sqrt{\left[\right(x_2-x_1{)}^2+(y_2-y_1{)}^2]}$

= $\sqrt{\left[\right(-2-3{)}^2+(5-(-7\left){)}^2\right]}$

= $\sqrt{\left[\right(-5{)}^2+\left(12{)}^2\right]}$

= $\sqrt{[25+144]}$

= $\sqrt{169}$

= $13$

Hence the radius is $13$ units.

(ii) Given $AB$ is the diameter and $C$ is the centre of the circle.

$\therefore$ By midpoint formula, $-2=(x+3)/2$

$\Rightarrow -4=x+3$

$\Rightarrow x=-4-3=-7$

$\therefore$ By midpoint formula, $5=(-7+y)/2$

$\Rightarrow 10=-7+y$

$\Rightarrow y=10+7=17$

Hence the co-ordinates of $B$ are $(-7,17)$.