First slide

Definition of a circle

Question

If the lines $2 x + 3 y + 1 = 0$ and $3 x - y - 4 = 0$
lie along diameters of a circle of circumference $10 π,$ then the equation of the circle is

Moderate

A

$x^{2} + y^{2} + 2 x - 2 y - 23 = 0$
B

$x^{2} + y^{2} - 2 x - 2 y - 23 = 0$
C

$x^{2} + y^{2} + 2 x + 2 y - 23 = 0$
D

$x^{2} + y^{2} - 2 x + 2 y - 23 = 0$

Solution

Lines $2 x + 3 y + 1 = 0$ and $3 x - y - 4 = 0$ intersect at (1, -1).
So, the coordinates of the centre of the circle are
(1, -1). Let r be the radius of the circle. Then,
Circumference $= 10 π \Rightarrow 2 π r = 10 π \Rightarrow r = 5$
Hence, the equation of the circle is
$(x - 1)^{2} + (y + 1)^{2} = 5^{2}$ or, $x^{2} + y^{2} - 2 x + 2 y - 23 = 0$

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