Question

The equation of circle which passes through the origin and cuts off intercepts 5 and 6 from the positive parts of the axes respectively, is \( \left(x - \frac{5}{2}\right)^2 + (y - 3)^2 = \lambda \), where \( \lambda \) is

• A. \( \frac{61}{4} \)
• B. \( \frac{6}{4} \)
• C. \( \frac{1}{4} \)
• D. \( 0 \)

Hint:

When circle passes through a point, substitute it into the equation to find radius or constant.

Answer 1

The Correct Option is A

To find the value of \( \lambda \) in the equation of the circle, let's analyze the given information step-by-step:

1. The equation of the circle is given as: \(\left(x - \frac{5}{2}\right)^2 + (y - 3)^2 = \lambda\)
2. The circle passes through the origin (0,0). Substituting \(x = 0\) and \(y = 0\) into the circle's equation: \(\left(0 - \frac{5}{2}\right)^2 + (0 - 3)^2 = \lambda\)
3. Calculating separately:
   • \(\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\)
   • \((-3)^2 = 9\)
4. Adding these results: \(\lambda = \frac{25}{4} + 9 = \frac{25}{4} + \frac{36}{4} = \frac{61}{4}\)
5. Since \(\lambda = \frac{61}{4}\), this matches option:
   • \(\frac{61}{4}\)

The intercepts provided (5 and 6 from the x-axis and y-axis, respectively) fit this calculation. Thus, the entire solution aligns perfectly with the given conditions, confirming the validity of the result.