Angle between tangents drawn to circle ${ x }^{ 2 }+{ y }^{ 2 }=20$, from the point $\left( 6,2 \right) $ is
- $\dfrac { \pi }{ 2 } $
- $\pi $
- $\dfrac { \pi }{ 4 } $
- $2\pi $
The point (6, 2) is outside the circle x^2 + y^2 = 20 (since 36 + 4 = 40 > 20). The distance from the center (0,0) to (6,2) is sqrt(40). The radius is sqrt(20). Let theta be the angle between the tangent and the line connecting the center to the point. sin(theta) = r/d = sqrt(20)/sqrt(40) = 1/sqrt(2). So theta = 45 degrees. The total angle between the two tangents is 2 * theta = 90 degrees, which is pi/2.
The given circle is x squared plus y squared equals 20, so its radius r is the square root of 20 and its center is (0, 0). The distance d from the external point (6, 2) to the center is the square root of (6 squared plus 2 squared), which equals the square root of 40. Using the property that the angle between two tangents is 2 arcsin(r/d), we get 2 arcsin(square root of 20 divided by square root of 40), simplifying to 2 arcsin(1 divided by the square root of 2). This evaluates to 2 multiplied by pi divided by 4, giving the angle as pi divided by 2.