Question

what is the equation of a line through the center of (x - 3)^2+(x + 2)^2 = 25 parallel to the line 7x+5y=-9

Explanation:

Step1: Find the center of the circle

The standard - form of a circle equation is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center. For the circle $(x - 3)^2+(y + 2)^2 = 25$, the center is $(3,-2)$.

Step2: Find the slope of the given line

Rewrite the line $7x+5y=-9$ in slope - intercept form $y=mx + c$, where $m$ is the slope. We get $5y=-7x - 9$, so $y=-\frac{7}{5}x-\frac{9}{5}$. The slope $m =-\frac{7}{5}$.

Step3: Use the point - slope form to find the line equation

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope. Using the point $(3,-2)$ and $m =-\frac{7}{5}$, we have $y+2=-\frac{7}{5}(x - 3)$.
Expand it: $y+2=-\frac{7}{5}x+\frac{21}{5}$.
Move all terms to one side: $7x+5y=21 - 10$.

Answer:

$7x + 5y=11$