Calculate the length of a tangent to the circle: x^2+y^2+8x−2y−8=0 at its point of tangency from the point P(2,4)
+128474 See the graph here :
This is a circle with a radius of 5 centered at (-4,1)
The distance from this center to (2.4) is a hypotenuse of a right triangle.......the radius of the circle forms one leg and the distance from (2,4) to the point of tangency is the other leg
Distance^2 from (-4,1) to (2,4) = (2 + 4)^2 + ( 4 - 1)^2 = 36 + 9 = 45
So...using the Pythagorean Theorem, the length^2 of the other leg is
45 - 5^2 = 45 - 25 = 20
And the sqrt of this is the distance from (2,4) to the point of tangency = sqrt (20) = 2sqrt (5)
Here's a graph :
