Question

Drag the tiles to the boxes to form correct pairs. Not all tiles will be used. Match the circle equations in general form with their corresponding equations in standard form. x2 + y2 − 4x + 12y − 20 = 0
(x − 6)2 + (y − 4)2 = 56
x2 + y2 + 6x − 8y − 10 = 0
(x − 2)2 + (y + 6)2 = 60
3x2 + 3y2 + 12x + 18y − 15 = 0
(x + 2)2 + (y + 3)2 = 18
5x2 + 5y2 − 10x + 20y − 30 = 0
(x + 1)2 + (y − 6)2 = 46
2x2 + 2y2 − 24x − 16y − 8 = 0
x2 + y2 + 2x − 12y − 9 = 0

Answer 1

Answer:

1) For $x^2 + y^2 - 4x + 12y - 20 = 0$, the standard form is $(x-2)^2 + (y+6)^2 = 60$

2) For $x^2 + y^2 + 6x - 8y - 10 = 0$, the standard form is $(x + 3)^2 + (y - 4)^2 = 35$

3) For $3x^2 + 3y^2 + 12x + 18y - 15 = 0$,  the standard form is $(x + 2)^2 + (y+ 3)^2 = 18$

4) For $5x^2 + 5y^2 - 10x + 20y - 30 = 0$,  the standard form is $(x - 1)^2 + (y+ 2)^2 = 11$

5) For $2x^2 + 2y^2 - 24x - 16y - 8 = 0$,  the standard form is $(x - 6)^2 + (y+ 4)^2 = 56$

6) For$x^2 + y^2 + 2x - 12y - 9 = 0$, the standard form is $(x+1)^2 + (y-6)^2 = 46$

Step-by-step explanation:

This can be done using the completing the square method.

The standard equation of a circle is given by $(x - a)^2 + (y-b)^2 = r^2$

1) For $x^2 + y^2 - 4x + 12y - 20 = 0$

$x^2 - 4x + y^2 + 12y = 20\\x^2 - 4x + 2^2 + y^2 + 12y + 6^2 = 20 + 4 + 36\\(x-2)^2 + (y+6)^2 = 60$

Therefore, for $x^2 + y^2 - 4x + 12y - 20 = 0$, the standard form is $(x-2)^2 + (y+6)^2 = 60$

2) For $x^2 + y^2 + 6x - 8y - 10 = 0$

$x^2 + 6x + y^2 - 8y = 10\\x^2 + 6x + 3^2 + y^2 - 8y + 4^2 = 10 + 9 + 16\\(x + 3)^2 + (y- 4)^2 = 35$

Therefore, for $x^2 + y^2 + 6x - 8y - 10 = 0$, the standard form is $(x + 3)^2 + (y - 4)^2 = 35$

3)  For $3x^2 + 3y^2 + 12x + 18y - 15 = 0$

Divide through by 3

$x^2 + y^2 + 4x + 6y = 5$

$x^2 + y^2 + 4x + 6y = 5\\x^2 + 4x + 2^2 + y^2 + 6y + 3^2 = 5 + 4 + 9\\(x + 2)^2 + (y+ 3)^2 = 18$

Therefore, for $3x^2 + 3y^2 + 12x + 18y - 15 = 0$,  the standard form is $(x + 2)^2 + (y+ 3)^2 = 18$

4)  For $5x^2 + 5y^2 - 10x + 20y - 30 = 0$

Divide through by 5

$x^2 + y^2 - 2x + 4y = 6$

$x^2 + y^2 -2x + 4y = 6\\x^2 - 2x + 1^2 + y^2 + 4y + 2^2 = 6 + 1 + 4\\(x - 1)^2 + (y+ 2)^2 = 11$

Therefore, for $5x^2 + 5y^2 - 10x + 20y - 30 = 0$,  the standard form is $(x - 1)^2 + (y+ 2)^2 = 11$

5) For $2x^2 + 2y^2 - 24x - 16y - 8 = 0$

Divide through by 2

$x^2 + y^2 - 12x - 8y = 4$

$x^2 + y^2 - 12x - 8y = 4\\x^2 - 12x + 6^2 + y^2 - 8y + 4^2 = 4 + 36 + 16\\(x - 6)^2 + (y+ 4)^2 = 56$

Therefore, for $2x^2 + 2y^2 - 24x - 16y - 8 = 0$,  the standard form is $(x - 6)^2 + (y+ 4)^2 = 56$

6) For $x^2 + y^2 + 2x - 12y - 9 = 0$

$x^2 + 2x + y^2 - 12y = 9\\x^2 + 2x + 1^2 + y^2 - 12y + 6^2 = 9 + 1 + 36\\(x+1)^2 + (y-6)^2 = 46$

Therefore, for$x^2 + y^2 + 2x - 12y - 9 = 0$, the standard form is $(x+1)^2 + (y-6)^2 = 46$