Question

find the centre and the radius of the circle 2(x+1)*2+2y*2=25

Answer 1

Answer:

Center: #(3,2)color(white)("XXXXX")#Radius: #5#
#color(white)("XXX")#(see below for method)

Explanation:

An equation in the form:
#color(white)("XXX")(x-color(red)a)^2+(y-color(blue)b)^2=color(magenta)r^2#
is the standard form for the equation of a circle with center #(color(red)a,color(blue)b)# and radius #color(magenta)r#.

Lets try to convert the given equation:
#color(white)("XXX")x^2+y^2-6x-4y-12=0#
into the standard form for the equation of a circle.

Group the #x# and #y# terms separately and "move" the constant to the right side of the equation:
#color(white)("XXX")x^2-6xcolor(white)(+3^2)+y^2-4ycolor(white)(+2^2)=12#

Complete the square for each of #x# and #y#
#color(white)("XXX")x^2-6xcolor(magenta)(+3^2)+y^2-4ycolor(orange)(+2^2)=12color(magenta)(+3^2)color(orange)(+2^2)#

Write the left side as the sum of two squared binomials
and simplify the result on the right side
#color(white)("XXX")(x-3)^2+(y-2)^2=25#

Express the right side as a square.
#color(white)("XXX")(x-color(red)3)^2+(y-color(blue)2)^2=color(green)5^2#
...the equation for a circle with center #(color(red)3,color(blue)2)# and radius #color(green)5#