Question

Write an equation for the translation of x ^ 2 + y ^ 2 = 25 by 6 units left and 3 units down.

Answer 1

ANSWER

The equation after the translation is
${(x + 6)}^{2} + {(y - 3)}^{2} = 25$

EXPLANATION

The given equation is
${x}^{2} + {y}^{2} = 25.$

This is an equation of a circle , centered at the origin and with radius
$5 \: units.$

If this circle is translated 6 units to left, then the x-coordinate of the centre will now be at
$x = - 6.$

Also, if the circle is translated 3 units up, then the y-coordinate of the centre will now be at
$y = 3$

The new centre is now,

$(-6,3).$

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

${(x - a)}^{2} + {(y - b)}^{2} = {r}^{2}$

Where

$a=-6,b=3,r=5$

The equation now becomes,

${(x - - 6)}^{2} + {(y - 3)}^{2} = {5}^{2}$

${(x + 6)}^{2} + {(y - 3)}^{2} = 25$

Answer 2

The equation would be: (x + 6)^2 + (y + 3)^2 = 25

This equation will produce a circle. If you add the numbers inside the parenthesis to be squared, you can center of the circle. Thus, translating the shape.