Finding the Equation of Circle Using Line and Circle Intercept

This problem involves finding the Equation of Circle where the diameter is determined by the intersection points of a given line and a given circle. The line is $y = x$, and the circle is $x^2 + y^2 - 2x = 0$. The intercept points on the line $y=x$ by the circle form the segment AB, which is the diameter of the required circle.

Steps to Find the Equation of Circle

We need to follow these steps to determine the required Equation of Circle:

1. Find the points of intersection between the line and the circle.
2. Use these intersection points as the endpoints of the diameter to find the Equation of Circle.

Step 1: Find the Intersection Points

We are given the line $y = x$ and the circle $x^2 + y^2 - 2x = 0$. To find the intersection points, we substitute the equation of the line into the equation of the circle:

Substitute $y = x$ into $x^2 + y^2 - 2x = 0$:

$\qquad x^2 + (x)^2 - 2x = 0$

$\qquad x^2 + x^2 - 2x = 0$

$\qquad 2x^2 - 2x = 0$

Now, we factor the equation to find the values of $x$:

$\qquad 2x(x - 1) = 0$

This gives us two possible values for $x$:

• $2x = 0 \implies x = 0$
• $x - 1 = 0 \implies x = 1$

Since $y = x$, the corresponding $y$ values are:

• When $x = 0$, $y = 0$. So, one intersection point is $(0, 0)$.
• When $x = 1$, $y = 1$. So, the other intersection point is $(1, 1)$.

These two points, $(0, 0)$ and $(1, 1)$, are the endpoints of the diameter AB.

Step 2: Find the Equation of Circle with AB as Diameter

If the endpoints of a diameter of a circle are $(x_1, y_1)$ and $(x_2, y_2)$, the equation of the circle is given by the diameter form formula:

$\qquad (x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$

Here, the endpoints of the diameter are $(x_1, y_1) = (0, 0)$ and $(x_2, y_2) = (1, 1)$. Substitute these coordinates into the formula:

$\qquad (x - 0)(x - 1) + (y - 0)(y - 1) = 0$

$\qquad x(x - 1) + y(y - 1) = 0$

Expand the terms:

$\qquad x^2 - x + y^2 - y = 0$

Rearrange the terms to get the standard form of the Equation of Circle:

$\qquad x^2 + y^2 - x - y = 0$

This is the required Equation of Circle with AB as the diameter, formed by the intercept of the line $y=x$ and the circle $x^2 + y^2 - 2x = 0$. This problem is a classic example of using the diameter form of the Equation of Circle after finding the intersection points of a line and a circle.