Expert: Ahmed Salami - 1/5/2010

Question
9y^2-25x^2-400x-1825=0

i need to identify what conic section it is then solve all the problems to that specific conic section.

(i.e. if it is a parabola i would need to know the vertex, axis, focus, directrix, what direction it would go (up, down, left, right))


please help me if you can, and thank you :)

if you could please show me how to do it and everything?
thank you.

Answer
Hi Sara,
The general form of any conic section is Ax² + Bxy + Cy² + Dx + Ey + F = 0
In most cases, B = 0 because any other value would cause the conic to be rotated and makes analysis much more difficult. The general equation can be algebraically manipulated to become the standard form of the specific conic it describes but it would first be useful to know what that conic is. There are a number of ways to determine the type of conic section an equation represents but we would follow the easiest way here simply by comparing A and C.
If A = C, it describes a circle.
If either A or C (but not both) equals 0, the equation describes a parabola.
If A and C are both positive or both negative (but not equal), it is an ellipse.
If A and C have opposite signs, it is a hyperbola.

Rewriting your equation, we have
-25x² + 9y² - 400x - 1825 = 0
or
25x² - 9y² + 400x + 1825 = 0
and it can be clearly seen that this is a hyperbola.

Now, the basic equation of a hyperbola is written in the standard form
x²/a² - y²/b² = 1
OR
x²/b² - y²/a² = -1
depending on the particular equation.
And in special cases we could have
(x - h)²/a² - (y - k)²/b² = 1
OR
(x - h)²/b² - (y - k)²/a² = -1

Rearranging 25x² - 9y² + 400x + 1825 = 0,
dividing by 25,
x² - 9y²/25 + 16x + 73 = 0
x² + 16x - 9y²/25 + 73 = 0
completing squares,
(x + 8)² - 64 - 9y²/25 + 73 = 0
(x + 8)² - 9y²/25 + 9 = 0
(x + 8)² - 9y²/25 = -9
dividing by 9
(x + 8)²/9 - y²/25 = -1
(x + 8)²/3² - y²/5² = -1

The standard hyperbola x²/b² - y²/a² = -1 has foci at (0,ae) and (0,-ae) where e is the eccentricity given by
e = √[1 + (b²/a²)]
and it opens up and down.
The directrices are the lines y = a/e and y = -a/e
The vertices are at (0,a) and (0,-a)
The semi-major axis is the y axis (x = 0) and the semi-minor axis is the x axis (y = 0).

For the special case of (x - h)²/b² - (y - k)²/a² = -1
The foci are at (h,k + ae) and (h,k - ae)
The directrices are the lines y = k + a/e and y = k - a/e
The vertices are at (h,k + a) and (h,k - a)
The semi-major axis is the line x = h and the semi-minor axis is the line y = k.

Comparing the hyperbola (x + 8)²/3² - y²/5² = -1
a = 5, b = 3, h = -8, k = 0
e = √[1 + (b²/a²)]
 = √[1 + (3²/5²)]
 = √1.36
 = 1.17

I'm sure you can complete the rest.
You can always get back to me.

Regards