The point P is on circle $\displaystyle { x }^{ 2 }+{ y }^{ 2 }=36$ and tangents are drawn to the circle $\displaystyle { x }^{ 2 }+{ y }^{ 2 }=18$ from P. The angle between the tangent lines is
The point P is on x^2 + y^2 = 36 (radius 6). The tangents are to x^2 + y^2 = 18 (radius sqrt(18)). In the right triangle formed by the center, the point of tangency, and P, the hypotenuse is 6 and one leg is sqrt(18). sin(theta/2) = sqrt(18)/6 = sqrt(2)/2. Thus theta/2 = 45 degrees, and theta = 90 degrees.
The point P lies on the circle with radius 6, so its distance from the origin is 6. Tangents are drawn to the circle with radius the square root of 18, which simplifies to 3 times the square root of 2. Using the right triangle formula relating the tangent length, the external point's distance, and the circle radius, the square of the tangent length equals 6 squared minus (3 times the square root of 2) squared, which is 36 minus 18, or 18. The angle between the two tangents satisfies the equation sin(half the angle) equals the radius divided by the distance to the center, so sin(alpha) equals (3 times the square root of 2) divided by 6, making alpha 45 degrees. Therefore, the full angle between the tangents is 2 times 45 degrees, which is 90 degrees.