Multiple choice

The length of tangent from point $(5, 1)$ to the circle $\displaystyle { x }^{ 2 }+{ y }^{ 2 }+6x-4y-3=0$ is:

  1. 81

  2. 29

  3. 7

  4. 21
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C Correct answer
Explanation

The length of the tangent from (x1, y1) to the circle x^2 + y^2 + 2gx + 2fy + c = 0 is sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c). Substituting (5, 1) into x^2 + y^2 + 6x - 4y - 3 gives sqrt(25 + 1 + 30 - 4 - 3) = sqrt(49) = 7.

AI explanation

The length of a tangent from an external point is found using the power of a point formula. Substitute x equals 5 and y equals 1 into the circle's expression to get 5 squared plus 1 squared plus 6 times 5 minus 4 times 1 minus 3. This simplifies to 25 plus 1 plus 30 minus 4 minus 3, equaling 49. The tangent length is the square root of this value, so the length is 7.

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