Q90 / 180 · COMEDK 2016 · Mathematics
The equation of the line which is tangent to both the circle $x^{2}+y^{2}=5$ and the parabola $y^{2}=40 x$ is
- Correct answerA$2 x+y+5=0$
- B$2 x-y-5=0$
- C$2 x-y+5=0$
- D$2 x-y \pm 5=0$
Solution
Let the equation of line by $y=m x+C$
Since, this is the tangent to the circle $x^{2}+y^{2}=5$
So,
$\begin{aligned}
C &=\pm a \sqrt{1+m^{2}} \\
&=\pm \sqrt{5} \sqrt{1+m^{2}} ...(i)
\end{aligned}$
Also, the above line is tangent to the parabola
$y^{2}=40 x$
So,
$c=\frac{a}{m}=\frac{10}{m}...(ii)$
By Eqs. (i) and (ii), we get
$\begin{aligned} \frac{10}{m} &=\pm \sqrt{5} \sqrt{1+m^{2}} \\ m^{4}+m^{2}-20 &=0 \\\left(m^{2}+5\right)\left(m^{2}-4\right) &=0 \\ m^{2} &=4\left(m^{2} \neq-5\right) \\ m &=\pm 2 \Rightarrow c=\pm 5 \\ \text { Here, } \quad y &=\pm 2 x \pm 5 \end{aligned}$
Hence, the correct answer is option: 'A'.