Final answer:
The center of the circle is at (-2, 3), not (4, -6), and the radius is 7 units, not 6 units or 49 units. Statements a and c are true, while b and d are false.
Step-by-step explanation:
The equation of a circle provided is x2 + y2 + 4x - 6y - 36 = 0. To convert this equation to the standard form, we need to complete the square for both the x and y terms.
To complete the square for the x terms: Group the x terms and half the coefficient of x, then add the square of that half to both sides. Similarly, for the y terms: group the y terms, half the coefficient of y, and add the square of that half to both sides. After adding these values, subtract the same on the other side to maintain equality. Once this is done for both x and y terms, the equation will be in the standard form of a circle (x - h)2 + (y - k)2 = r2, where (h,k) is the center and r is the radius.
Let's find the correct center and radius for the provided equation:
- Group x terms and y terms: (x2 + 4x) + (y2 - 6y) = 36
- Complete the square for x: (x + 2)2 - 4 + (y2 - 6y) = 36
- Complete the square for y: (x + 2)2 - 4 + (y - 3)2 - 9 = 36
- Combine constants on the right: (x + 2)2 + (y - 3)2 = 36 + 4 + 9
- (x + 2)2 + (y - 3)2 = 49
So, the center of the circle is at (-2, 3) and the radius of the circle is 7 units (since the square root of 49 is 7).
Hence, statement a is true, and statement c is false as the radius is not 6 but 7 units. Statement b is false as the center of the circle is not (4, -6). Statement d is also false as the radius of the circle is not 49 units, 49 is the square of the radius.