Question

A circle has a center (2, 4). prove that if the radius of the circle is less than 5, then the circle does not intersect the line y=x-6.

Answer 1

the distance from the center to the line is about 5.65, too far from a circle of radius 5.

Explanation:

We can prove this by looking at the distance from the center of the circle to the line.

The distance from point (x, y) to line ax+by+c=0 is given by ...

d = |ax +by +c|/√(a^2 +b^2)

For the line ...

x -y -6 = 0

and the point (2, 4), the distance is ...

d = |2 -4 -6|/(1^2 +(-1)^2) = |-8|/√2 = 8/√2 ≈ 5.65685

This tells us that a circle with radius 5.65 or less will not intersect the line. A circle of radius 5 is smaller than that. A circle with radius 5 will not intersect the line y = x-6.