Multiple choice

Tangents are drawn through the point $P(-2, -3)$ to touch the circle $x^{2}+y^{2}-2x-10y+1=0$. The length of line segment $PA$, $A$ being the point on the circle where the line meets the circle at coincident points, is

  1. $16$
  2. $4\sqrt{3}$
  3. $48$

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Length of tangent from (x1, y1) to circle x^2+y^2+2gx+2fy+c=0 is sqrt(x1^2+y1^2+2gx1+2fy1+c). Here, g=-1, f=-5, c=1. Point (-2, -3). Length = sqrt((-2)^2 + (-3)^2 - 2(-2) - 10(-3) + 1) = sqrt(4 + 9 + 4 + 30 + 1) = sqrt(48) = 4*sqrt(3).

AI explanation

The center of the circle x^2 + y^2 - 2x - 10y + 1 = 0 is (1, 5) and its radius r is sqrt(1^2 + 5^2 - 1) = 5. The distance d from the point P(-2, -3) to the center is sqrt((-2-1)^2 + (-3-5)^2) = sqrt(9 + 64) = sqrt(73). The length of the tangent from an external point is given by the formula L = sqrt(d^2 - r^2). Substituting the values gives L = sqrt((sqrt(73))^2 - 5^2) = sqrt(73 - 25) = sqrt(48) = 4*sqrt(3).

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