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. \(5\)

Hint:

Use product of slopes = -1 for perpendicular lines.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To find lines joining the origin to intersection points, we "homogenize" the curve equation using the line equation. Since the lines are perpendicular, the sum of coefficients of \(x^{2}\) and \(y^{2}\) must be zero.
Step 2: Detailed Explanation:
1. Line equation: \(y - 3x = c \implies \frac{y - 3x}{c} = 1\).
2. Homogenize the circle equation \(x^{2} + y^{2} = 4(1)^{2}\):
\[ x^{2} + y^{2} = 4 \left( \frac{y - 3x}{c} \right)^{2} \]
\[ c^{2}(x^{2} + y^{2}) = 4(y^{2} + 9x^{2} - 6xy) \]
\[ c^{2}x^{2} + c^{2}y^{2} = 36x^{2} + 4y^{2} - 24xy \]
\[ (c^{2} - 36)x^{2} + 24xy + (c^{2} - 4)y^{2} = 0 \]
3. For right angle (\(90^{\circ}\)), Coefficient of \(x^{2} + \) Coefficient of \(y^{2} = 0\):
\[ (c^{2} - 36) + (c^{2} - 4) = 0 \]
\[ 2c^{2} - 40 = 0 \implies c^{2} = 20 \]
Step 3: Final Answer:
The value of \(c^{2}\) is 20.