Question

The intercept of a line $y = x$ by the circle ${x^2} + {y^2} - 2x = 0$ is AB. What is the equation of the circle with AB as a diameter?
${\text{A}}{\text{. }}{x^2} + {y^2} + x + y = 0$
${\text{B}}{\text{. }}{x^2} + {y^2} - x - y = 0$
${\text{C}}{\text{. }}{x^2} + {y^2} + x - y = 0$
${\text{D}}{\text{. }}$None of these

Answer 1

Here, we will use the general equation of the circle passing through the points of intersection of any circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ and any line $ax + by + d = 0$ which is ${x^2} + {y^2} + 2gx + 2fy + c + \lambda \left( {ax + by + d = 0} \right) = 0$.

Complete step-by-step answer:
Given, equation of the circle is ${x^2} + {y^2} - 2x = 0$ and equation of line is $y = x \Rightarrow x - y = 0$
As we know that the equation of the circle passing through the point of intersection of a circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ and a straight line $ax + by + d = 0$ is given by
${x^2} + {y^2} + 2gx + 2fy + c + \lambda \left( {ax + by + d = 0} \right) = 0$

Using above concept, we can say that the required equation of the circle is
${x^2} + {y^2} - 2x + \lambda \left( {x - y} \right) = 0{\text{ }} \to {\text{(1)}}$

In the above equation of circle, $\lambda $ is the unknown whose value is needed.

Also, we know that general equation of a circle with center coordinate as C$\left( {{x_1},{y_1}} \right)$ and radius as $r$ is given by ${\left( {x - {x_1}} \right)^2} + {\left( {y - {y_1}} \right)^2} = {r^2}{\text{ }} \to {\text{(2)}}$

Now we will convert equation (1) in the same form as equation (2), we get
\[{x^2} + {y^2} - 2x + \lambda x - \lambda y = 0 \Rightarrow {x^2} + x\left( {\lambda - 2} \right) + {y^2} - \lambda y = 0\]

Factoring the above equation with the help of completing the square method
\[
   \Rightarrow {x^2} + x\left( {\lambda - 2} \right) + {\left[ {\dfrac{{\lambda - 2}}{2}} \right]^2} + {y^2} - \lambda y + {\left( {\dfrac{\lambda }{2}} \right)^2} - {\left[ {\dfrac{{\lambda - 2}}{2}} \right]^2} - {\left( {\dfrac{\lambda }{2}} \right)^2} = 0 \Rightarrow {\left[ {x + \left( {\dfrac{{\lambda - 2}}{2}} \right)} \right]^2} + {\left[ {y - \left( {\dfrac{\lambda }{2}} \right)} \right]^2} = \dfrac{{{{\left( {\lambda - 2} \right)}^2}}}{4} + \dfrac{{{\lambda ^2}}}{4} \\
   \Rightarrow {\left[ {x - \left( { - \left( {\dfrac{{\lambda - 2}}{2}} \right)} \right)} \right]^2} + {\left[ {y - \left( {\dfrac{\lambda }{2}} \right)} \right]^2} = \dfrac{{{\lambda ^2} + 4 - 4\lambda + {\lambda ^2}}}{4} \Rightarrow {\left[ {x - \left( {\dfrac{{2 - \lambda }}{2}} \right)} \right]^2} + {\left[ {y - \left( {\dfrac{\lambda }{2}} \right)} \right]^2} = \dfrac{{2\left( {{\lambda ^2} - 2\lambda + 2} \right)}}{4} \\
   \Rightarrow {\left[ {x - \left( {\dfrac{{2 - \lambda }}{2}} \right)} \right]^2} + {\left[ {y - \left( {\dfrac{\lambda }{2}} \right)} \right]^2} = \dfrac{{{\lambda ^2} - 2\lambda + 2}}{2}{\text{ }} \to {\text{(3)}} \\
 \]

On comparing equation (3) with equation (2), we can write
Center Coordinate of the required circle is C$\left[ {\left( {\dfrac{{2 - \lambda }}{2}} \right),\dfrac{\lambda }{2}} \right]$.

Since the required circle has AB as the diameter or we can say that the center coordinate of the required circle should lie on the given straight line whose equation is $y = x$.
i.e., Put $x = \left( {\dfrac{{2 - \lambda }}{2}} \right)$ and $y = \left( {\dfrac{\lambda }{2}} \right)$ in equation $y = x$ since the center coordinates satisfies the equation of the straight line.
\[ \Rightarrow x = y \Rightarrow \left( {\dfrac{{2 - \lambda }}{2}} \right) = \left( {\dfrac{\lambda }{2}} \right) \Rightarrow 2 - \lambda = \lambda \Rightarrow 2\lambda = 2 \Rightarrow \lambda = 1\]

Now let us put the value of $\lambda $ in equation (1), we get
$
  {x^2} + {y^2} - 2x + 1\left( {x - y} \right) = 0 \Rightarrow {x^2} + {y^2} - 2x + x - y = 0{\text{ }} \\
   \Rightarrow {x^2} + {y^2} - x - y = 0 \\
 $

The above equation corresponds to the required equation of the circle with AB as diameter.

Therefore, option B is correct.

Note- In these types of problems, an unknown usually exists which can be evaluated by simply comparing the equation of the circle having an unknown obtained with the general form of the circle. In this way, center coordinates are obtained and then the given condition is satisfied which will eventually give us the value of the unknown.