Question

A pair of tangents are drawn from the origin to the circle \( x^2 + y^2 + 20(x + y) + 20 = 0 \), then the equation of the pair of tangent are

• A. \( x^2 + y^2 - 5xy = 0 \)
• B. \( x^2 + y^2 + 2xy = 0 \)
• C. \( x^2 + y^2 - 2xy = 0 \)
• D. \( 2x^2 + 2y^2 + 5xy = 0 \)

Hint:

To find the equation of tangents from the origin to a circle, use the general formula and complete the square for the circle equation.

Answer 1

The Correct Option is D

Step 1: Rewrite the equation of the circle.
The given equation of the circle is: \[ x^2 + y^2 + 20(x + y) + 20 = 0 \] Completing the square, we get the standard form of the circle equation.
Step 2: Find the equation of the tangents.
Using the formula for tangents from the origin to the circle, we get the equation \( 2x^2 + 2y^2 + 5xy = 0 \). Final Answer: \[ \boxed{2x^2 + 2y^2 + 5xy = 0} \]