Question

The condition for a line $y=2x+c$ to touch the circle $x^2+y^2=16$ is

• A. $c=10$
• B. $c^2=80$
• C. $c=12$
• D. $c^2=64$

Answer 1

The correct option is B

$c^2=80$

Explanation for the correct option:

Compute the required value.

Given the line equation: $y=2x+c$ and circle equation $x^2+y^2=16$

If $y=2x+c$ is a tangent,

then $r=d=\sqrt{16}=4$,

Since, $x^2+y^2=r^2$ is the circle equation, where r is the radius of the circle, centered at origin $C(0,0)$.

As radius is the perpendicular distance from center to line, we have

$$\begin{gathered} d=\frac{\left|2x-y+c\right|}{\sqrt{1^2+2^2}}\text{at}\left(0,0\right) \\ 4=\frac{\left|0-0+c\right|}{\sqrt{1^2+2^2}} \end{gathered}$$

On calculating, we get

$$\begin{gathered} \text{c}=4\times \sqrt{5} \\ {c}^2=16\times 5 \\ =80 \\ {c}^2=80 \end{gathered}$$

Hence, option (B) is correct answer.