Question

select the correct answer. 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 form

The standard form of a circle 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.

Step2: Test 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 check other options for confirmation.

Step3: Test 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: Test Option C: \((0,13)\)

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

Step5: Test 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. \((20,-3)\)