Multiple choice

Passage

Tangents are drawn from the point $P(1,-1)$ to the circle $x^{2}+y^{2}-4x-6y-3=0$ with centre C. A and B are the points of contact.

The equation of the circle circumscribing the triangle formed by pair of tangent and corresponding chord of contact, is

  1. $x^{2}+y^{2}+3x-2y-1=0$
  2. $x^{2}+y^{2}+3x+2y-1=0$
  3. $x^{2}+y^{2}-3x-2y-1=0$

  4. none
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle circumscribing the triangle formed by the pair of tangents and the chord of contact is the circle with the line segment CP as diameter, where C is the center of the circle and P is the external point. C = (2, 3), P = (1, -1). Diameter form: (x - 2)(x - 1) + (y - 3)(y + 1) = 0. x^2 - 3x + 2 + y^2 - 2y - 3 = 0. x^2 + y^2 - 3x - 2y - 1 = 0.

AI explanation

The center of the given circle is C(2, 3). The required circle circumscribing the triangle formed by the pair of tangents from P(1, -1) and the chord of contact AB has the line segment CP as its diameter. Using the diameter form of a circle, the equation is (x - 1)(x - 2) + (y + 1)(y - 3) = 0. Expanding and simplifying this expression yields x squared plus y squared minus 3x minus 2y minus 1 equals 0.

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