WAEC 2018 · Paper 2 · Q9

Coordinate geometry & circles

A circle is drawn through the points $(3, 2)$ , $(-1, -2)$ and $(5, -4)$ . Find the:

(a)
coordinates of the centre of the circle;
Separate values with commas, e.g. 3, −2
(b)
radius of the circle;
(c)
equation of the circle.
Show the answer
$x^2 + y^2 - 5x + 3y - 4 = 0$

Try it on a graph

The three points and the circle through them.

Worked solution (try it first)

Substitute each point into $x^2 + y^2 + 2gx + 2fy + k = 0$ .
$(3, 2)$ : $6g + 4f + k = -13$ .
$(-1, -2)$ : $-2g - 4f + k = -5$ .
$(5, -4)$ : $10g - 8f + k = -41$ .
First minus second: $8g + 8f = -8$ , so $g + f = -1$ .
Third minus first: $4g - 12f = -28$ , so $g - 3f = -7$ .
Subtract: $4f = 6$ , so $f = \frac32$ and $g = -\frac52$ .
Then $k = -13 + 15 - 6 = -4$ .

(a)

The centre is $(-g, -f) = \left(\frac52, -\frac32\right)$ .

(b)

$r = \sqrt{g^2 + f^2 - k}$
$= \sqrt{\frac{25}{4} + \frac94 + 4}$
$= \sqrt{12.5}$
$\approx 3.536$ .

(c)

$x^2 + y^2 - 5x + 3y - 4 = 0$ .

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