4x^2-25y^2-8x-100y-196=0 how do i find the conic section to find the center?

complete the square

4x^2-25y^2-8x-100y-196=0
4(x^2 - 2x + ....) - 25(y^2 + 4y + ....) = 196
4(x^2 - 2x + 1) - 25(y^2 + 4y + 4....) = 196 + 4 + 100</b)
4(x-1)^2 - 25(y+2)^2 = 300

(x-1)^2/75 - (y+2)^2/12 = 1

do you recognize this conic?
Can you determine the centre?

x^2+y ^2+ 4x + 6y - 12 = 0 how do i find the domain

are you the same person as bbb ?

Do not switch names.
Start a new post for a new question..

Try completing the square like I just showed you, from there you should be able to find the centre and the radius.

The domain is easy after that

To find the conic section and the center of the given equation, 4x^2 - 25y^2 - 8x - 100y - 196 = 0, we first need to write the equation in its standard form. The standard form of a conic section equation takes one of several possible forms, depending on the type of conic section (ellipse, hyperbola, or parabola).

In this case, let's complete the square to convert the equation into the appropriate standard form for identifying the conic section.

Step 1: Group the x-terms and y-terms separately:
4x^2 - 8x - 25y^2 - 100y = 196

Step 2: Complete the square for the x-terms by factoring out the common coefficient (4):
4(x^2 - 2x) - 25y^2 - 100y = 196

Step 3: To complete the square for the x-terms, take half the coefficient of x (-2), square it (-2^2 = 4), and add it to both sides of the equation:
4(x^2 - 2x + 1) - 25y^2 - 100y = 196 + 4

Step 4: Simplify the x-terms:
4(x - 1)^2 - 25y^2 - 100y = 200

Step 5: Repeat steps 2 to 4 for the y-terms. Start by factoring out the common coefficient (-25):
4(x - 1)^2 - 25(y^2 + 4y) = 200

Step 6: To complete the square for the y-terms, take half the coefficient of y (4), square it (4^2 = 16), and add it to both sides of the equation:
4(x - 1)^2 - 25(y^2 + 4y + 4) = 200 + 16

Step 7: Simplify the y-terms:
4(x - 1)^2 - 25(y + 2)^2 = 216

Step 8: Divide both sides by the constant term to isolate the variables on one side of the equation:
(x - 1)^2/54 - (y + 2)^2/ 216 = 1

Now that the equation is in the standard form (with the variable squared terms on different sides and equal to 1), we can identify the conic section.

Looking at the equation, we see that the x^2 term and y^2 term have different coefficients, and the difference between these coefficients is positive. This indicates that the given equation represents a hyperbola.

To find the center of the hyperbola, we can identify the values (h, k) where the equation is in the form:

(x - h)^2/a^2 - (y - k)^2/b^2 = 1

In our equation, we have:

(x - 1)^2/54 - (y + 2)^2/216 = 1

Comparing this to the standard form, we can see that the center of the hyperbola is at the point (h, k) = (1, -2).