Question

How do you write an equation for the translation of x^2 + y^2 = 25 by 7 units left and 2 units down?

Answer 1

The equation of the translated circle is:
`(x + 7)^2 + (y + 2)^2 = 25`.

Explanation:

The standard form of a circle is
`(x -h)^2 + (y - k)^2 = r^2`,
where `r` is the radius of the circle and `(h, k)` is the center of the circle.

In `x^2 + y^2 = 25`, the center is at `(0, 0)`. When the circle is translated left 7 units, it means that the new center has `h = -7`. Translating the circle down 2 units means that the new center has `k = -2`. So, the translated circle has the equation

`(x + 7)^2 + (y + 2)^2 = 25`.