First slide

Definition of a circle

Question

If $2 x + 3 y - 6 = 0$ and $9 x + 6 y - 18 = 0$ cuts the axes in concyclic points, then the centre of the circle, is

Moderate

A

(2, 3)
B

(3, 2)
C

(5, 5)
D

(5/2, 5/2)

Solution

The equation of the circle passing through the points of intersection of the lines $2 x + 3 y - 6 = 0$ and $9 x + 6 y - 18 = 0$ with the coordinate axes is
$\begin{array}{l} (2 x + 3 y - 6) (9 x + 6 y - 18) - (2 \times 6 + 9 \times 3) x y = 0 \\ \Rightarrow x^{2} + y^{2} - 5 x - 5 y + 6 = 0 \end{array}$
The coordinates of the centre are $(5 / 2, 5 / 2)$ .

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