Question

If the line $(3x+ay-20=0)$ cuts the circle $(x^2+y^2)=25$ at real, distinct or coincident points, then $a$ belongs to the interval

• A. $[-\sqrt{7},\sqrt{7}]$
• B. $(-\sqrt{7},\sqrt{7})$
• C. $(-\infty ,-\sqrt{7}]\cup [\sqrt{7},\infty )$
• D. None of these

Answer 1

Answer: (C)

$(-\infty ,-\sqrt{7}]\cup [\sqrt{7},\infty )$

The length of the $\perp$ from the centre $(0,0)$ of the given circle to the line $3x+ay-20=0$ is $\frac{|3\left(0\right)+a\left(0\right)-20|}{\sqrt{9+a^2}}=\frac{20}{\sqrt{9+a^2}}.$

Radius of the given circle $=5$

Since the line cuts the circle at real, distinct or coincident points

$\therefore \frac{20}{\sqrt{9+a^2}}\le 5\Rightarrow a^2+9\ge 16\Rightarrow a^2-7\ge 0$

$\Rightarrow (a+\sqrt{7})(a-\sqrt{7})\ge 0$

$\Rightarrow a\in (-\infty ,-\sqrt{7}]\cup [\sqrt{7},\infty ).$