First slide

Definition of a circle

Question

The lines $2 x - 3 y - 5 = 0$ and $3 x - 4 y = 7$ are
diameters of a circle of area $154 (= 49 π)$ sq. units, then the equation of
the circle is

Moderate

A

$x^{2} + y^{2} + 2 x - 2 y - 62 = 0$
B

$x^{2} + y^{2} + 2 x - 2 y - 47 = 0$
C

$x^{2} + y^{2} - 2 x + 2 y - 47 = 0$
D

$x^{2} + y^{2} - 2 x + 2 y - 62 = 0$

Solution

The centre of the required circle lies at the
intersection of $2 x - 3 y - 5 = 0$ and $3 x - 4 y - 7 = 0 .$ Thus, the
coordinates of the centre are (1, -1).
Let r be the radius of the circle. Then, by hypothesis, we have
$π r^{2} = 154 \Rightarrow \frac{22}{7} r^{2} = 154 \Rightarrow r = 7$
Hence, the equation of the required circle is
$(x - 1)^{2} + (y + 1)^{2} = 7^{2} \Rightarrow x^{2} + y^{2} - 2 x + 2 y - 47 = 0$

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