Multiple choice

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

  1. $3$
  2. $4$
  3. $0$

  4. None of these
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A 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). Here, x1=2, y1=5, g=-1, f=-1.5, c=-1. Length = sqrt(2^2 + 5^2 - 2(2) - 3(5) - 1) = sqrt(4 + 25 - 4 - 15 - 1) = sqrt(9) = 3.

AI explanation

The length of a tangent drawn from an external point (x1, y1) to a circle given by the equation x^2 + y^2 + 2gx + 2fy + c = 0 is found using the formula sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c). For the given circle x^2 + y^2 - 2x - 3y - 1 = 0, the values are g = -1, f = -3/2, and c = -1. Substituting the point (2, 5) into the formula gives sqrt((2)^2 + (5)^2 - 2(2) - 3(5) - 1). This simplifies to sqrt(4 + 25 - 4 - 15 - 1) = sqrt(9), which equals 3.

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