Math Snap

STEP 1

1. The given equation is $25x^2 - 9y^2 - 250x - 36y + 364 = 0$.
2. The equation represents a hyperbola.
3. The goal is to rewrite the equation in the standard form of a hyperbola.

STEP 2

1. Rearrange the equation to group $x$ and $y$ terms.
2. Complete the square for both $x$ and $y$ terms.
3. Simplify and write the equation in standard form.

STEP 3

Rearrange the equation to group $x$ and $y$ terms:
$$25x^2 - 250x - 9y^2 - 36y = -364$$

STEP 4

Complete the square for the $x$ terms:
1. Factor out $25$ from the $x$ terms:
$$25(x^2 - 10x)$$ 2. Complete the square:
$$x^2 - 10x$$ $$= (x - 5)^2 - 25$$ 3. Substitute back:
$$25((x - 5)^2 - 25) = 25(x - 5)^2 - 625$$

STEP 5

Complete the square for the $y$ terms:
1. Factor out $-9$ from the $y$ terms:
$$-9(y^2 + 4y)$$ 2. Complete the square:
$$y^2 + 4y$$ $$= (y + 2)^2 - 4$$ 3. Substitute back:
$$-9((y + 2)^2 - 4) = -9(y + 2)^2 + 36$$

Substitute completed squares back into the equation and simplify:
$$25(x - 5)^2 - 625 - 9(y + 2)^2 + 36 = -364$$ Combine constants:
$$25(x - 5)^2 - 9(y + 2)^2 - 589 = -364$$ Add $589$ to both sides:
$$25(x - 5)^2 - 9(y + 2)^2 = 225$$ Divide the entire equation by $225$ to get the standard form:
$$\frac{(x - 5)^2}{9} - \frac{(y + 2)^2}{25} = 1$$ The equation in conic form is:
$$\boxed{\frac{(x - 5)^2}{9} - \frac{(y + 2)^2}{25} = 1}$$