Certainly! Let's solve the problem step-by-step.
We start with the original equation of the circle:
[tex]\[ x^2 + y^2 = 25 \][/tex]
This represents a circle centered at (0, 0) with a radius of 5.
Step 1: Translating the Circle to the Right
When we translate a circle to the right by a number of units, we modify the [tex]\(x\)[/tex]-coordinate by subtracting the translation distance from [tex]\(x\)[/tex]. Translating two units to the right involves modifying the [tex]\(x\)[/tex] term to [tex]\((x - 2)\)[/tex]:
[tex]\[ (x - 2)^2 + y^2 = 25 \][/tex]
Step 2: Translating the Circle Upward
When we translate a circle upward by a number of units, we modify the [tex]\(y\)[/tex]-coordinate by subtracting the translation distance from [tex]\(y\)[/tex]. Translating four units up involves modifying the [tex]\(y\)[/tex] term to [tex]\((y - 4)\)[/tex]:
[tex]\[ (x - 2)^2 + (y - 4)^2 = 25 \][/tex]
Therefore, the new equation for the translated circle is:
[tex]\[ (x-2)^2 + (y-4)^2 = 25 \][/tex]
So, the correct answer is:
[tex]\[
\boxed{(x-2)^2+(y-4)^2=25}
\][/tex]
