Multiple choice

Find the angle between the two tangents from the origin to the circle(x-7)2+(y+1)2=25.

  1. pi/2

  2. pi/3

  3. pi/4

  4. pi/5
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A Correct answer
Explanation

The distance from the origin (0,0) to the center (7,-1) is sqrt(7^2 + (-1)^2) = sqrt(50). The radius is 5. The angle between the tangents is 2 * arcsin(r/d) = 2 * arcsin(5/sqrt(50)) = 2 * arcsin(1/sqrt(2)) = 2 * 45 degrees = 90 degrees, which is pi/2.

AI explanation

The given circle has center (7, -1) and radius 5. The distance from the origin (0, 0) to the center is calculated as the square root of 7 squared plus negative 1 squared, which equals the square root of 50, or 5 times the square root of 2. Since the distance from the origin to the center is exactly 5 times the square root of 2, the origin lies outside the circle, and the length of each tangent from the origin is the square root of 50 minus 25, which equals 5. The triangle formed by the two radii and the two tangents is an isosceles right triangle, making the angle between the two tangents equal to pi divided by 2.

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