Conics - Two problems I'm having trouble on

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Hi there,

I'm currently working on Conics and need help with the following two problems.


01) Identify the type of conic and write the equation in standard form.

9x[sup:1ovtbciq]2[/sup:1ovtbciq] + 4y[sup:1ovtbciq]2[/sup:1ovtbciq] - 18x + 16y - 11 = 0

I used the determinant, B[sup:1ovtbciq]2[/sup:1ovtbciq] - 4AC, and know this is an ellipse.

Afterwards, I moved the loose number, factor, and complete the square to get:

9(x[sup:1ovtbciq]2[/sup:1ovtbciq] - 2x + 1) + 4(y[sup:1ovtbciq]2[/sup:1ovtbciq] + 4y + 4) = 11 + 9 + 16

9(x - 1)[sup:1ovtbciq]2[/sup:1ovtbciq] + 4(y + 2)[sup:1ovtbciq]2[/sup:1ovtbciq] = 36

(x - 1)[sup:1ovtbciq]2[/sup:1ovtbciq]/4 + (y + 2)[sup:1ovtbciq]2[/sup:1ovtbciq]/9 = 1

The standard form is: (y + 2)[sup:1ovtbciq]2[/sup:1ovtbciq]/9 + (x - 1)[sup:1ovtbciq]2[/sup:1ovtbciq]/4 = 1

Is that the correct answer? If not, where did I go wrong at?

A side question related to the problem above. Suppose there are coefficients in front of (x - 1)[sup:1ovtbciq]2[/sup:1ovtbciq] and (y + 2)[sup:1ovtbciq]2[/sup:1ovtbciq] and I'm looking for the vertices and foci, how exactly should I go about finding that? I know how to find it when there are no coefficients in front, but as for when there are, what should I do?

02) Prove that x[sup:1ovtbciq]2[/sup:1ovtbciq] + 4xy + 4y[sup:1ovtbciq]2[/sup:1ovtbciq] - 30x - 90y + 450 = 0 is the equation of a parabola by rotating the coordinate axes through a suitable angle a.

I have no idea where to start for this. I know I need to get B = 0 by eliminating the 4xy term, using:

cot 2a = (A - C) / B.

I'm also given cos 2a = -3/5.

I'm racking my brain on all the identities I studied, and I cannot seem to get how cos 2a = -3/5 and how to get there. Is it linked to the: B cos 2a + (C - A) sin 2a?

EDIT: Never mind for the second question. I figure I needed to use the Pythagorean Theorem and the Half-Angle Identities for this problem. But I would really appreciate feedback on Problem 01. Did I do it right?

Any help is great, thank you very much.
Abby

 

[royhaas]

You could start with \(\displaystyle \cot^2(\theta) = cos^2(\theta)/(1-cos^2(\theta))\).

 

[fasteddie65]

If you know cos 2a, use Half-Angle identities. cos (a/2) = ?((1 + cos a)/2)

Since cos 2a = -3/5, cos a = ?((1 - 3/5)/2 = ?((2/5)/2) = ?(1/5) = 1/?5. Since sin^2 a + cos^2 a = 1, sin^2 a = 4/5, so sin a = 2/?5.