Multiple choice

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

  1. $7$
  2. $49$
  3. $63$
  4. $21$
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A Correct answer
Explanation

The length of the tangent from (x1, y1) to a circle is sqrt(S1), where S1 is the result of plugging the point into the circle equation. S1 = 5^2 + 1^2 + 6(5) - 4(1) - 3 = 25 + 1 + 30 - 4 - 3 = 49. The length is sqrt(49) = 7.

AI explanation

Substitute the external point (5, 1) into the circle's equation to find the power of the point, which gives the square of the tangent length. The calculation is 5 squared plus 1 squared plus 6 times 5 minus 4 times 1 minus 3, equaling 25 plus 1 plus 30 minus 4 minus 3, which equals 49. Taking the square root of 49 gives the length of the tangent as 7.

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