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વર્તૂળ દ્વારા રેખા પર બનાવેલ અંત:ખંડ $AB$ હોય તો $AB$ જેનો વ્યાસ હોય તેવા વર્તૂળનું સમીકરણ મેળવો.
${x^2} + {y^2} - x - y = 0$
${x^2} + {y^2} - 2x - y = 0$
${x^2} + {y^2} - x + y = 0$
${x^2} + {y^2} + x - y = 0$
Solution
(a) Equation of any circle passing through the point of intersection of ${x^2} + {y^2} – 2x = 0$ and $y = x$ is
${x^2} + {y^2} – 2x + \lambda (y – x) = 0$
or ${x^2} + {y^2} – (2 + \lambda )x + \lambda y = 0$
Its center is $\left( {\frac{{2 + \lambda }}{2},\;\frac{{ – \lambda }}{2}} \right)$.
For $AB$ to be the diameter of the required circle, the centre must lie on $AB$
$i.e.$, $\frac{{2 + \lambda }}{2} = – \frac{\lambda }{2} $
$\Rightarrow \lambda = – 1$.
Thus the required equation of the circle is
${x^2} + {y^2} – 2x – y + x = 0$ or ${x^2} + {y^2} – x – y = 0$.
