Multiple choice

The square of the length of the tangent from $\left( 3,-4 \right) $ to the circle ${ x }^{ 2 }+{ y }^{ 2 }-4x-6y+3=0$ is

  1. 20

  2. 30

  3. 40

  4. 50
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C Correct answer
Explanation

The length of the tangent from (x1, y1) to x^2 + y^2 + 2gx + 2fy + c = 0 is sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c). Here, x1=3, y1=-4, g=-2, f=-3, c=3. Length^2 = 3^2 + (-4)^2 - 4(3) - 6(-4) + 3 = 9 + 16 - 12 + 24 + 3 = 40.

AI explanation

To find the square of the tangent length, use the power of a point formula, which requires the center and radius of the circle derived from its equation. Completing the square for the equation x squared plus y squared minus 4x minus 6y plus 3 equals 0 gives the center at (2, 3) and a radius of the square root of 10. The square of the tangent length equals the square of the distance from the point (3, -4) to the center minus the square of the radius. The distance squared is (3 minus 2) squared plus (-4 minus 3) squared, which equals 1 plus 49 equals 50. Subtracting the radius squared gives 50 minus 10 equals 40, which is the required value.

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