Question

What is the minimum distance between any point on the circle x square+y square=25 and
the line y=−3/4x+75/4?

Answer 1

We will use a line perpendicular to line y=−3/4x+75/4. As we are working with a circle with it's center at the origin the line will go through the origin. That will be y=4/3x. If you multiply there slopes you get -1 which is how you determine 2 perpendicular lines.

You can now calculate where they intersect by using the 2 equations:
y = 4/3 x
y= −3/4x+75/4

That means 4/3 x = −3/4x+75/4.
i.e. x = 9.

If you plug that into the first equation you get y = 4/3 (9) = 12.

Using pythagoras the distance from the origin to the intersection of teh 2 lines are the square root of (9 ^2 + 12^2 ) = 15.

So the distance from the circle is 15-radius = 15-5 = 10.