Find the equation of the circle with a radius of 4 and its center at the intersection of $4 x + y = 2$ and $- 3 x -$ $4 y = 5$.
A) $\quad ( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
B) $\quad ( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 4$
C) $( x - 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 16$
D) $( x + 1 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 4$
E) none of these

Question

Quizplus · Accepted Answer

First, we need to find the center of the circle. To do this, we solve the system of equations:

$$4x + y = 2$$
$$-3x - 4y = 5$$

Multiplying the first equation by 4, we get:

$$16x + 4y = 8$$

Adding this to the second equation, we get:

$$13x = 13$$

So, $x=1$. Substituting this value into the first equation, we get $y=-2$. Therefore, the center of the circle is $(1,-2)$.

Next, we use the formula for the equation of a circle with center $(h,k)$ and radius $r$:

$$(x-h)^2 + (y-k)^2 = r^2$$

Plugging in the values we know, we get:

$$(x-1)^2 + (y+2)^2 = 16$$

Which is answer choice C.

Find the Equation of the Circle with a Radius of 4