Question

17 what is the equation for the translation of ( x^2 + y^2 = 25 ) two units to the left and four units down?
( circ (x - 2)^2 + (y + 4)^2 = 25 )
( circ (x + 2)^2 + (y + 4)^2 = 25 )
( circ (x - 2)^2 + (y - 4)^2 = 25 )
( circ (x + 2)^2 + (y - 4)^2 = 25 )

Explanation:

Step1: Recall circle standard form

The standard equation of a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h,k)$ is the center and $r$ is the radius. For $x^2 + y^2 = 25$, the center is $(0,0)$ and $r=5$.

Step2: Apply horizontal translation

Translating 2 units left means subtracting 2 from the $x$-coordinate of the center: $h = 0 - 2 = -2$. Substitute into the $x$-term: $(x - (-2))^2=(x + 2)^2$.

Step3: Apply vertical translation

Translating 4 units down means subtracting 4 from the $y$-coordinate of the center: $k = 0 - 4 = -4$. Substitute into the $y$-term: $(y - (-4))^2=(y + 4)^2$.

Step4: Write final equation

Keep the radius squared $r^2=25$. The translated equation is $(x + 2)^2 + (y + 4)^2 = 25$.

Answer:

$\boldsymbol{(x + 2)^2 + (y + 4)^2 = 25}$