Question

Which of the following is the equation for a circle with a radius of r and center at (h, v)? A. (x + h)2 + (y + v)2 = r2 B. h2 + v2 = r2 C. (x - h)2 + (y - v)2 = r2 D. (x - v)2 + (y - h)2 = r2

Answer 1

Answer:

$c) (x-h)^{2}+(y-v)^{2}=r^{2}$

Step-by-step explanation:

Hi There,

1. The Circle formula in its standard form is given by:

$(x-a)^{2}+(y-b)^{2}=r^{2}\Rightarrow C=(a,b)$

2) This coordinates a, b  are the Center coordinates of the Center, a point distant from the circumference by the radius.

3) Because we can find derive this formula from that. (Check the graph below)

There's a point P(x,y) whose distance to C(h,v) is the radius, we need to calculate it numerically:

$r=\sqrt{(x-h)^{2}+(y-v)^{2}} \:or\:r^{2}=(x-h)^{2}+(y-v)^{2}\\r=\sqrt{x^{2}-2hx+h^{2}+y^{2}+2vy+v^{2}}\\r=\sqrt{x^{2}+h^{2}+y^{2}+v^{2}}\\(r)^{2}=(\sqrt{x^{2}+h^{2}+y^{2}+v^{2}})^{2}\\r^{2}=x^{2}+h^{2}+y^{2}+v^{2}\\r^{2}=x^{2}+y^{2}+h^{2}+v^{2}$

4) Hence, as the Center has its coordinates C(h,v) then its circle formula is:

$c) (x-h)^{2}+(y-v)^{2}=r^{2}$

Answer 2

(x - h)^2 + (y - v)^2 = r^2.

The answer would be C.

Hope this helps! :)