Answer:
See below
Explanation:
We solve the equations as a system of simultaneous equations to find the point of intersection. The point of intersection is the point of tangency.
x - 2y = 10
x² + y² = 20
x = 2y + 10
(2y + 10)² + y² = 20
x = 2y + 10
4y² + 40y + 100 + y² = 20
x = 2y + 10
5y² + 40y + 80 = 0
x = 2y + 10
y² + 8y + 16 = 0
x = 2y + 10
(y + 4)² = 0
x = 2(-4) + 10
y =-4
x = 2
y = -4
The point of tangency is (2, -4).
The equation of the circle is x² + y² = 0. The center of the circle is (0, 0). The point of tangency is (2, -4). The line in which the radius to the point of tangency lies has slope
m = (-4 - 0)/(2 - 0) = -2
The given line has equation
x - 2y = 10
-2y = -x + 10
y = (1/2)x - 5,
so its slope is 1/2.
The radius of a circle drawn to the point of tangency and the tangent are perpendicular.
The slopes are -2 and 1/2, respectively, which are negative reciprocals, so they are perpendicular. Therefore, line x - 2y = 10 is a tangent to circle x² + y² = 20.