Question

If 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 -

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

Answer 1

Answer: (A)

20

Answer 2

Solution:

Let the two points of intersection of the circle

$$x^2+y^2=4$$

and the line

$$y=3x+c$$

be $A$ and $B$.

The chord $AB$ subtends the right angle $\angle AOB = 90^\circ$ at the origin $O$. For a circle of radius $R$ with a chord subtending an angle $\theta$ at the centre, the chord length is

$$AB = 2R\sin\frac{\theta}{2}.$$

Here, $R=2$ and $\theta = 90^\circ$, so

$$AB = 2 \times 2 \sin(45^\circ)=4 \times \frac{\sqrt2}{2}=2\sqrt2.$$

On the other hand, the perpendicular distance $d$ from the centre $O$ to the line $y=3x+c$ is given by

$$d = \frac{|c|}{\sqrt{1+3^2}}=\frac{|c|}{\sqrt{10}}.$$

The length of the chord in the circle from a line at distance $d$ is also

$$AB = 2\sqrt{R^2-d^2}=2\sqrt{4-\frac{c^2}{10}}.$$

Equate the two expressions for $AB$:

$$2\sqrt{4-\frac{c^2}{10}} = 2\sqrt2.$$

Divide both sides by 2:

$$\sqrt{4-\frac{c^2}{10}}=\sqrt2.$$

Square both sides:

$$4-\frac{c^2}{10}= 2.$$

Rearrange:

$$\frac{c^2}{10}=4-2=2 \quad\Longrightarrow\quad c^2=20.$$

Thus, the correct option is (A) 20.

---

Explanation (minimal):

1. Compute chord length using the formula $2R\sin(\theta/2)$ for $\theta=90^\circ$.

2. Express chord length as $2\sqrt{4-\frac{c^2}{10}}$ using distance from origin to line.

3. Equate and solve to obtain $c^2=20$.