The line $3x+5y+9=0$ w.r.t the circle $x^{2}+y^{2}-4x+6y+5=0$ is
- chord dividing circumference in $1:3$ ratio
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diameter
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tangent
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outside line
Reveal answer
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B
Correct answer
Explanation
The center of the circle x^2 + y^2 - 4x + 6y + 5 = 0 is (2, -3). Substituting these coordinates into the line equation 3x + 5y + 9 = 0 gives 3(2) + 5(-3) + 9 = 6 - 15 + 9 = 0. Since the center lies on the line, the line is a diameter.
AI explanation
The center of the circle x^2+y^2-4x+6y+5=0 is found by taking the negative average of the x and y coefficients, giving center (2, -3). Plugging this point into the line equation 3x+5y+9=0 yields 3(2) + 5(-3) + 9 = 0. Because the center of the circle lies directly on the line, the line must be a diameter.