Write 36x^2-360x-25y^2-100y=100 in standard form. Then state whether the graphs of the equation is a parabola, circle, ellipse, or hyperbola.
I started with:
36(x^2 -10x+25)-25(y^2+4y+4)=100+36(25)-25(4)
which leads to: 36(x^2-10x+25)/900 - 25(y^2+4y+4)/900=1
Simplified to: (x-5)^2/25 - (y+2)^2/36; ellipse.
Is this correct?
Yes.
Algebra is correct, but this is then a hyperbola.
Also, try to solve more complicated problems containing a term proportional to xy, e.g.:
x^2 + 3 x + y^2 - 12 y + 15 x y = 1588
Yes, it's a hyperbola. I overlooked that - sign.
Your steps are almost correct, but there seems to be a slight mistake in your simplification.
Starting with:
36(x^2 -10x+25)-25(y^2+4y+4)=100+36(25)-25(4)
You correctly expanded the equation:
36(x^2 -10x+25)-25(y^2+4y+4) = 900 - 100
To simplify further, you need to divide the entire equation by the right-hand side, which is 900 - 100. This will make the right-hand side equal to 1:
36(x^2 -10x+25)/(900 - 100) - 25(y^2+4y+4)/(900 - 100) = 1
Simplifying the expression inside the parentheses:
36(x^2 -10x+25)/800 - 25(y^2+4y+4)/800 = 1
Factoring the trinomials:
36(x - 5)^2/800 - 25(y + 2)^2/800 = 1
Now, let's simplify the coefficients further by dividing both sides of the equation by the common factor of 8:
(9/200)(x - 5)^2 - (25/200)(y + 2)^2 = 1
Finally, we can rewrite the equation in standard form by multiplying both sides by -1, so that the leading coefficients are positive:
(25/200)(y + 2)^2 - (9/200)(x - 5)^2 = -1
Therefore, the standard form of the equation is:
(25/200)(y + 2)^2 - (9/200)(x - 5)^2 = -1
The graph of this equation is an ellipse.