The pairs of equations that represent concentric circles are:
[tex]i) ~3x^2 + 3y^2 + 12x - 6y -21 = 0\\\\~~~~~5x^2 + 5y^2 - 30x + 20y - 10 = 0\\\\ii)~5x^2 + 5y^2 - 10x + 40y - 75 = 0\\\\~~~~~x^2 + y^2 - 2x + 8y - 13 = 0\\\\iii)~5x^2 + 5y^2 - 30x + 20y - 10 = 0\\\\~~~~~3x^2 + 3y^2 - 18x + 12y - 81 = 0\\\\iv)~2x^2 + 2y^2 - 8x + 12y - 40 = 0\\\\~~~~~4x^2 + 4y^2 - 16x + 24y - 28 = 0[/tex]
"The circles which have the same center and different radii are called concentric circles. "
The general equation of the circle is, [tex]x^2+y^2+2gx+2fy+c=0[/tex] where (-g, -f) is the center of the circle and radius = [tex]\sqrt{g^2+f^2-c}[/tex]
For given question,
We have been given the pairs of equations.
We know, the general equation of circles is,
[tex]x^2 + y^2 + 2gx + 2fy +c = 0[/tex]
We need to find the pairs of equations that represent concentric circles.
1) Consider the equation [tex]3x^2 + 3y^2 + 12x - 6y -21 = 0[/tex]
Divide above equation by 3,
[tex]x^2 + y^2 + 4x - 2y -7 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g,-f) = (-2, 1)
2) Consider the equation [tex]5x^2 + 5y^2 - 10x + 40y - 75 = 0[/tex]
Divide above equation by 5,
[tex]x^2 + y^2 - 2x + 8y - 15 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f) = (1, - 4)
3) Consider an equation, [tex]5x^2 + 5y^2 - 30x + 20y - 10 = 0[/tex]
Divide above equation by 5,
[tex]\Rightarrow x^2 + y^2 - 6x + 4y - 2 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f ) = (3, -2)
4) Consider an equation, [tex]4x^2 + 4y^2 + 16x - 8y - 308 = 0[/tex]
Divide above equation by 4,
[tex]\Rightarrow x^2 + y^2 + 4x - 2y - 77 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f ) = (-2, 1)
5) Consider an equation, [tex]x^2 + y^2 - 12x - 8y - 100 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f ) = (6, 4)
6) Consider an equation, [tex]2x^2 + 2y^2 - 8x + 12y - 40 = 0[/tex]
Divide above equation by 2,
[tex]\Rightarrow x^2 + y^2 - 4x + 6y - 20 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f ) = (2, -3)
7) Consider an equation, [tex]4x^2 + 4y^2 - 16x + 24y - 28 = 0[/tex]
Divide above equation by 4,
[tex]\Rightarrow x^2 + y^2 - 4x + 6y - 7 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f ) = (2, -3)
8) Consider an equation, [tex]3x^2 + 3y^2 - 18x + 12y - 81 = 0[/tex]
Divide above equation by 3,
[tex]\Rightarrow x^2 + y^2 - 6x + 4y - 27 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f ) = (3, -2)
9) Consider an equation, [tex]x^2 + y^2 - 2x + 8y - 13 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f ) = (1, -4)
10) Consider an equation, [tex]x^2 + y^2 + 24x + 30y + 17 = 0[/tex]
By comparing this equation with the general equation of circle, the center of above circle is (-g, -f ) = (-12, -15)
Therefore, the pairs of equations that represent concentric circles are:
[tex]i) ~3x^2 + 3y^2 + 12x - 6y -21 = 0\\\\~~~~~5x^2 + 5y^2 - 30x + 20y - 10 = 0\\\\ii)~5x^2 + 5y^2 - 10x + 40y - 75 = 0\\\\~~~~~x^2 + y^2 - 2x + 8y - 13 = 0\\\\iii)~5x^2 + 5y^2 - 30x + 20y - 10 = 0\\\\~~~~~3x^2 + 3y^2 - 18x + 12y - 81 = 0\\\\iv)~2x^2 + 2y^2 - 8x + 12y - 40 = 0\\\\~~~~~4x^2 + 4y^2 - 16x + 24y - 28 = 0[/tex]
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