SATHEE: Circle 4 Question 14

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

$\begin{aligned} (x^{2} + y^{2} - 2 x) + λ (y - x) & = 0 \\ \Rightarrow & x^{2} + y^{2} - (2 + λ) x + λ y & = 0 \end{aligned}$

Its centre is $\frac{2 + λ}{2}, \frac{- λ}{2}$ .

$\Rightarrow$ For $A B$ to be the diameter of the required circle the centre must be on $A B$ , i.e.

$\begin{aligned} 2 + λ & = - λ \\ \Rightarrow & λ & = - 1 \end{aligned}$

Therefore, equation of the required circle is

$\begin{aligned} x^{2} + y^{2} - (2 - 1) x - 1 \cdot y & = 0 \\ \Rightarrow & x^{2} + y^{2} - x - y & = 0 \end{aligned}$

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Circle 4 Question 14

14. The intercept on the line y=x by the circle x2+y2−2x=0 is AB. Equation of the circle with AB as a diameter is… .

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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… .