Question

Write the equation in standard form 45y^2 - 320x^2 + 6 = 2886.

Write the standard form of 9x^2 + 36x + 25y^2 - 50y = 164

I will give a medal AND fan if you help me!

Answer 1

I'm assuming I'm dealing with hyperbolas/ellipses/circles and the lies.

For the 1st one:
Let's start by taking away that 6.
45y^2 - 320x^2 = 2880

We have Ay^2 - Bx^2 = constant (where A and B are constants), which tells me we are dealing with a hyperbola. The standard form for a hyperbola has a 1 on the right side, so we divide by 2880.

We now have:
(y^2)/64 - (x^2)/9 = 1.

Done.

For the second one:
Factor out a constant from the x and y terms.
So we have:
9(x^2 + 4x) + 25(y^2 - 50x) = 164
This is an ellipse.
For these type of problems you always want to express the y and x terms in standard form as:
(y +/- some number)^2 and (x +/ some number)^2 respectively.
But wait! As you've noticed, we can't do this for the x and y terms we have.

So what now?
We can add some number to x^2 + 4 and y^2 - 50 such that it'll be able to factor them out into (x +/- some constant)^2 and (y +/- some constant)^2. We can do this as long as we add the constants to the right-hand side (For example, let's take x =4. Let's add 2 to it. x + 2 = 6. Statement is still valid and means the exact same thing. That is exactly what we're doing here).

So what is this number we're going to be adding?
We take the number in front of the x/y that's to the 1st power, half it, and then square it.
For the x term, we get:
(4/2)^2 = 4

For the y term, we get:
(50/2)^2 = 625.

But here's the tricky part. Once we add those numbers to their respective terms, we don't just add 4 and 625 To the right hand side. We have to add 36 and 15625. Why?

As you can see, the x^2 + .... and y^2 + ... terms are being multiplied by a constant. Since we're adding those numbers inside those parenthesis, we have to multiply the numbers by their respective constants and THEN add them to the right hand side.

So now we have:
9(x^2 + 4x + 4) + 25(y^2 - 50x + 625) = 15,825.

Divide both sides by 15,825 (right hand side in the end must always be 1!)
9(x^2 + 4x + 4)/15825 + (y^2 - 50x + 625)/633 = 1
(I put 9/15825 instead of the number you get from doing the division because I didn't get nice clean numbers for 9/15825).

Turn the quadratic terms (i.e x^2 +....) into the form (x + some constant)^2 (the constant can be negative)

(9/15825)(x+2)^2 + (1/633)(y - 25)^2 = 1

Answer 2

*typo 1st sentence. Lie should be likes.

Answer 3

edit again: I messed up on the y^2 term.
It should be y^2 -2 since I factored out 25.
So the constant we add is:
(2/2)^2 = 1.
So we instead get:
9(x + 2)^2 + 25(y^2 -2x + 1) = 225
So we get:
(x+2)^2/25 + (y - 1)^2/9 = 1
To go down even further:
(x+2)^2/5^2 + (y-1)^2/3^2 = 1

Answer 4

thank you!