Multiple choice

Passage

To the circle $ x^2+y^2=4$ two tangents are drawn from $P(4, 0)$, which touches the circle at $T_1$ and $T_2$ and a rhombus $PT_1P'T_2$ is completed.

Circumcentre of the triangle $PT_1T_2$ is at

  1. $(2, 0)$
  2. $(0, 2)$
  3. $\left (\displaystyle \frac{\sqrt{3}}{2},0 \right )$

  4. none of these
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A Correct answer
Explanation

The tangents from P(4,0) to x^2 + y^2 = 4 touch at points T1 and T2. The triangle PT1T2 is isosceles. The circumcentre of a triangle formed by tangents and the chord of contact is the midpoint of the line segment connecting the external point and the origin, or specifically calculated via geometry as (2,0).

AI explanation

The given circle x squared plus y squared equals 4 has center O at (0, 0) and radius r equals 2. In the rhombus P T1 P prime T2, the diagonal P P prime passes through the center O and is bisected by it, making the coordinates of P prime equal to (-4, 0). The points T1 and T2 lie on the circle, so angle P T1 P prime is 90 degrees, meaning T1 lies on the circle with P P prime as its diameter. The circle circumscribing triangle P T1 T2 is exactly this circle with diameter P P prime, having center at the midpoint of (4, 0) and (-4, 0), which is (2, 0).

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