Question

The angle between the pair of straight lines formed by joining the points of intersection of $x^2+y^2=4$and $y=3x+c$ to the origin is a right angle. Then, $c^2$ is equal to

• A. $20$
• B. $13$
• C. $\frac{1}{5}$
• D. $5$

Answer 1

The correct option is A

$20$

Find the value of $c^2$:

Given curves are $x^2+y^2=4$and $y=3x+c$

Step 1: On homogenizing both the curve:

$\Rightarrow \frac{y-3x}{c}=1,x^2+y^2=4{\left(1\right)}^2$

Step 2: Using the substitution method

$$\begin{gathered} x^2+y^2=4{\left(\frac{y-3x}{c}\right)}^2 \\ \Rightarrow c^2(x^2+y^2)=4(y^2-6xy+9x^2) \\ \Rightarrow x^2(c^2-36)+y^2(c^2-4)=-24xy \end{gathered}$$

We know that when pair of lines are at a right angle, then

Coefficient of $x^2$ $+$ coefficient of $y^2$$=0$

$$\begin{gathered} \Rightarrow c^2-36+c^2-4=0 \\ \Rightarrow c^2=20 \end{gathered}$$

Hence, option (A) is the correct answer.