Question

The number of integral values of \(\lambda\) for which \(x^2 + y^2\) +\(\lambda\) x + (1 - \(\lambda\))y + 5 = 0 is the equation of a circle whose radius does not exceed 5 is

• A. 14
• B. 15
• C. 16
• D. 18

Hint:

Use coefficient comparison to find centre and radius.

Answer 1

The Correct Option is C

Step 1: Compare with standard circle \[ (x-h)^2+(y-k)^2=r^2. \]
Step 2: Centre from coefficients: \[ h=-\lambda/2,\qquad k=-(1-\lambda)/2. \]
Step 3: Radius squared: \[ r^2=h^2+k^2-5 =\frac{\lambda^2+(1-\lambda)^2}{4}-5. \]
Step 4: Condition \(r\le5\Rightarrow r^2\le25\). Solve inequality gives feasible integers count 16. Hence → (C).