Circle 4 Question 14
14. The intercept on the line $y=x$ by the circle $x^{2}+y^{2}-2 x=0$ is $A B$. Equation of the circle with $A B$ as a diameter is… .
(1996, 1M)
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Answer:
Correct Answer: 14. $x^{2}+y^{2}-x-y=0$
Solution:
- Equation of any circle passing through the point of intersection of $x^{2}+y^{2}-2 x=0$ and $y=x$ is
$$ \begin{array}{rlrl} & & \left(x^{2}+y^{2}-2 x\right)+\lambda(y-x) & =0 \\ \Rightarrow & x^{2}+y^{2}-(2+\lambda) x+\lambda y & =0 \end{array} $$
Its centre is $\frac{2+\lambda}{2}, \frac{-\lambda}{2}$.
$\Rightarrow$ For $A B$ to be the diameter of the required circle the centre must be on $A B$, i.e.
$$ \begin{array}{rlrl} & & 2+\lambda & =-\lambda \\ \Rightarrow & \lambda & =-1 \end{array} $$
Therefore, equation of the required circle is
$$ \begin{aligned} & & x^{2}+y^{2}-(2-1) x-1 \cdot y & =0 \\ \Rightarrow & & x^{2}+y^{2}-x-y & =0 \end{aligned} $$
