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 Cannot Exceed $5$, Is

• A. $14$
• B. $16$
• C. $18$
• D. $20$

Answer 1

The correct option is B

$16$

Finding the Equation Of A Circle whose Radius Cannot Exceed 5 :

Option (A) $14$

Given in the question, the radius is less than or equal to 5

$$\begin{gathered} \sqrt{\left(\frac{{\lambda }^2}{4}+\frac{{(1-\lambda )}^2}{4}-5\right)}\le 5 \\ \Rightarrow {\lambda }^2+{(1-\lambda )}^2-20\le 100 \\ \Rightarrow 2{\lambda }^2-2\lambda -119\le 0 \\ \Rightarrow \frac{1-\sqrt{239}}{2}\le \lambda \le \frac{1-\sqrt{2392}}{2}-7.2\le \lambda \le 8.2 \end{gathered}$$

Therefore,

$\lambda =-7,-6-5.......7,8$

in all $16$ values.

Option A $16\ne 14$

Option A is wrong

Option (B) $16$ :

Therefore,

$\lambda =-7,-6-5.......7,8$

in all $16$ values.

Option B $16=16$

Hence option B is right answer.

Option (C) $18$:

Therefore,

$\lambda =-7,-6-5.......7,8$

in all $16$ values.

Option C $18\ne 16$

Option C is wrong

Option (D) $20$:

Therefore,

$\lambda =-7,-6-5.......7,8$

in all $16$ values.

Option D $16\ne 20$

Option D is wrong

Therefor option B is right answer.