Answer:
D
Step-by-step explanation:
We have the equation:
[tex]x^2+y^2+12x-10y-7=0[/tex]
And we would like to convert this to an equation for a circle.
This will require completing the square. Thus, let’s factor group our factors. We will also move the constant to the right. Therefore:
[tex](x^2+12x)+(y^2-10y)=7[/tex]
To complete the square, we divide the b term by 2, square it, and then add the resulting constant to both sides.
Therefore, for the first term, we will have 12/2. 12/2 is 6. 6 squared is 36.
Hence, we would add 36 to both sides.
For the second term, we have -10/2. -10/2 is 5. 5 squared is 25. Hence, we would add 25 to both sides.
This yields:
[tex](x^2+12x+36)+(y^2-10y+25)=7+36+25[/tex]
We can now factor the two terms using the perfect square trinomial pattern. We will also add on the right. Hence, our equation is:
[tex](x+6)^2+(y-5)^2=68[/tex]
Therefore, our answer is D.
