Question

If $y + 3x = 0$ is the equation of a chord of the circle, $x^2 + y^2 - 30x = 0$, then the equation of the circle with this chord as diameter is :

• A. $x^2 + y^2 + 3x + 9y = 0$
• B. $x^2 + y^2 - 3x + 9y = 0$
• C. $x^2 + y^2 - 3x - 9y = 0$
• D. $x^2 + y^2 + 3x - 9y = 0$

Answer 1

The Correct Option is D

To solve this problem, we need to find the equation of the circle for which the given chord \(y + 3x = 0\) is a diameter. The main steps are as follows:

1. Identify the given circle equation: The given circle has the equation \(x^2 + y^2 - 30x = 0\). To simplify, rewrite it as \(x^2 - 30x + y^2 = 0\). Completing the square for \(x\), we can express it as:
   • \(x^2 - 30x = (x - 15)^2 - 225\)
   • So, the equation becomes \((x - 15)^2 + y^2 = 225\).
2. Determine the endpoints of the chord: The chord equation is \(y + 3x = 0\), or \(y = -3x\). Let's find where it intersects the circle:
   • Substitute \(y = -3x\) into the circle's equation \((x - 15)^2 + (-3x)^2 = 225\)
   • Calculate: \((x - 15)^2 + 9x^2 = 225\)
   • Now expand and simplify: \(x^2 - 30x + 225 + 9x^2 = 225\)
   • Simplify to: \(10x^2 - 30x = 0\)
   • Factorize: \(10x(x - 3) = 0\). Hence, \(x = 0\) or \(x = 3\). For these x-values, \(y = 0\) or \(y = -9\).
3. Find the equation of the circle with this diameter:
   • The center of the circle is the midpoint of the diameter. Calculate the midpoint: \(\left(\frac{0+3}{2}, \frac{0+(-9)}{2}\right) = \left(\frac{3}{2}, -\frac{9}{2}\right)\)
   • Find the radius using the distance between these endpoints, which is the distance from the midpoint to either endpoint: \(r = \sqrt{\left(\frac{3}{2} - 0\right)^2 + \left(-\frac{9}{2} - 0\right)^2} = \sqrt{\left(\frac{3}{2}\right)^2 + \left(-\frac{9}{2}\right)^2} = \sqrt{\frac{9}{4} + \frac{81}{4}} = 5\)
   • The equation of the circle is: \(\left(x - \frac{3}{2}\right)^2 + \left(y + \frac{9}{2}\right)^2 = 25\)
   • After expanding it to remove fractions: \((2x - 3)^2 + (2y + 9)^2 = 100\)
   • Simplifies to: \(4x^2 - 12x + 9 + 4y^2 + 36y + 81 = 100\)
   • Simplify and subtract 100 from both sides: \(4x^2 + 4y^2 - 12x + 36y - 10 = 0\)
   • Divide through by 2 for simplicity: \(x^2 + y^2 - 3x + 9y = 0\)
4. Conclude with the answer: Hence, the equation of the circle with the given chord as its diameter is \(x^2 + y^2 + 3x - 9y = 0\).