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Find the combined equation of the pair of tangents drawn from the origin to the circle x² + y² + 4x + 6y + 9 = 0.

1. 3(x² + y²) = (x + 2y)²
2. 2(x² + y²) = (3x + y)²
3. 9(x² + y²) = (2x + 3y)²
4. x² + y² = (2x + 3y)²

Correct answer: 9(x² + y²) = (2x + 3y)²

Solution

Using SS1 = T²: S = x²+y²+4x+6y+9, S1 = 9 (at origin), T = 2x+3y+9. Expanding 9(x²+y²+4x+6y+9) = (2x+3y+9)² and simplifying gives 5x² - 12xy = 0. Separately, option C: 9(x²+y²) = (2x+3y)² expands to 9x²+9y² = 4x²+12xy+9y², giving 5x² = 12xy, i.e., x(5x-12y) = 0. Both are the same pair of lines: x = 0 and 5x - 12y = 0, confirming option C is correct.

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