ANSWER
The equation after the translation is
[tex] {(x + 6)}^{2} + {(y - 3)}^{2} = 25[/tex]
EXPLANATION
The given equation is
[tex] {x}^{2} + {y}^{2} = 25.[/tex]
This is an equation of a circle , centered at the origin and with radius
[tex]5 \: units.[/tex]
If this circle is translated 6 units to left, then the x-coordinate of the centre will now be at
[tex]x = - 6.[/tex]
Also, if the circle is translated 3 units up, then the y-coordinate of the centre will now be at
[tex]y = 3[/tex]
The new centre is now,
[tex](-6,3).[/tex]
The radius of the circle is not affected after the translation. It is still 5 units.
The new equation can be found using the formula
[tex] {(x - a)}^{2} + {(y - b)}^{2} = {r}^{2} [/tex]
Where
[tex]a=-6,b=3,r=5[/tex]
The equation now becomes,
[tex] {(x - - 6)}^{2} + {(y - 3)}^{2} = {5}^{2} [/tex]
[tex] {(x + 6)}^{2} + {(y - 3)}^{2} = 25[/tex]
