How do you find the equation that represents the image of circle #(x- 5)^2 + (y + 12)^2 = 169# after a translation 2 units right and 3 units down?

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Answer 1 · 2016-01-30T00:29:46

Adjust to standard form to see what the original centre is, translate the centre, then plug back into the equation to find:

#(x-7)^2+(y-(-15))^2 = 13^2#

or:

#(x-7)^2+(y+15)^2=169#

The standard equation of a circle with centre #(h,k)# and radius #r# is:

#(x-h)^2+(y-k)^2 = r^2#

With minor changes our starting equation is:

#(x-5)^2+(y-(-12))^2=13^2#

That is: It is a circle with centre #(5, -12)# and radius #13#.

If translated #2# units right and #3# units down then the centre will be #(5+2, -12-3) = (7, -15)# and the equation becomes:

#(x-7)^2+(y-(-15))^2 = 13^2#

or putting it back in similar form to the starting equation:

#(x-7)^2+(y+15)^2=169#