Question

The intercept on the line y = x by the circle x² + y² - 2x = 0 is AB. The equation of the circle with AB as diameter is ______.

पर्याय

• x² + y² + x + y = 0

• x² + y² − x − y = 0

• x² + y² − 3x + y = 0

• x² + y² + 3x − y = 0

Answer 1

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

Explanation:

Given circle:

x² + y² − 2x = 0

y = x

Step 1: Find intercept points (A and B)

Substitute y = x into the circle:

x² + x² − 2x = 0

2x² − 2x = 0

2x(x − 1) = 0

x = 0 or x = 1

So, A(0, 0) and B(1, 1)

Step 2: Equation of circle with AB as diameter.

If A(x₁, y₁) and B(x₂, y₂) are ends of diameter, equation of circle is:

(x − x₁​)(x − x₂​) + (y − y₁​)(y − y₂​) = 0

Substitute A(0, 0), B(1, 1)

(x − 0)(x − 1) + (y − 0)(y − 1) = 0

x(x − 1) + y(y − 1) = 0

x² − x + y² − y = 0

x² + y² − x − y = 0