Question

which point lies on the circle represented by the equation $x^2 + (y - 12)^2 = 25^2$?

a. $(20, -3)$

b. $(-7, 24)$

c. $(0, 13)$

d. $(-25, -13)$

Explanation:

Step1: Recall the circle equation formula

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius. For the given equation \(x^2+(y - 12)^2 = 25^2\), the center is \((0,12)\) and radius \(r = 25\). A point \((x,y)\) lies on the circle if it satisfies the equation. So we substitute each option into the equation.

Step2: Check option A \((20,-3)\)

Substitute \(x = 20\), \(y=-3\) into the equation:
Left - hand side (LHS) \(=20^2+(-3 - 12)^2=400+(-15)^2=400 + 225=625\)
Right - hand side (RHS) \(=25^2 = 625\). Since LHS = RHS, we can check other options for confirmation.

Step3: Check option B \((-7,24)\)

Substitute \(x=-7\), \(y = 24\) into the equation:
LHS \(=(-7)^2+(24 - 12)^2=49+12^2=49 + 144 = 193
eq625\)

Step4: Check option C \((0,13)\)

Substitute \(x = 0\), \(y=13\) into the equation:
LHS \(=0^2+(13 - 12)^2=0 + 1=1
eq625\)

Step5: Check option D \((-25,-13)\)

Substitute \(x=-25\), \(y=-13\) into the equation:
LHS \(=(-25)^2+(-13 - 12)^2=625+(-25)^2=625 + 625=1250
eq625\)

Answer:

A. \((20,-3)\)