The point (5,4) lies on a circle , what is the length of the radius of this circle if the centre is located at (3,2)?

Question

What is the length of the radius?

Answer 1

All points on the circle are joined to the centre by the radius.

r²=(5-3)²+(4-2)²=4+4=8, so radius r=√8=2/2 (=2.8284)

[(x-3)²+(y-2)²=8 is the equation of the circle]

Answer 2

The equation of the circle with radius 'r' and center at Origin (0, 0) is expressed as x²+y²=r²  … Eq.1.

When the center (0, 0) is moved to the point (3, 2), the equation of the new circle can be expressed as (x-3)²+(y-2)²=r²  … Eq.2

The point (5, 4) rests on the circumference of Eq.2, so Eq.2 can be rewritten plugging x=5 and y=4 into the equation.  (5-3)²+(4-2)²=r²

That is r²=2²+2²=8.  Here, we have the solution: r=8^½ =(4*2)^½=(4^½)*(2^½)=2*(2^½). We omitted (-)2*(2^½) because r>0.

The answer: The length of the radius is 2*(2^½).