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The question that is circled (2) is the question I need help with.

(https://i.stack.imgur.com/y3jMT.jpg)

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    This question is unclear. What do you supposed to do with this equation? – 2017-02-01
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    If you are going to simply post homework problems with no work, no context, and not even any effort to type out the problem here and merely give us a link we need not trust, I will tell you we do not accept such questions. – 2017-02-01
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    If you need help typing out the math, please see this guide on how to do so: [The MathJax guide](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) – 2017-02-01
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    "Conic form" can mean different things to different people. To some people, what you called "standard form" _is_ a conic form. It's much clearer if you write out the form that you want, for example $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,$ rather than merely naming it and hoping everyone else uses that name the _exact same way_ as your textbook. Mathematical language is relatively standardized, but not _that_ standardized. – 2017-02-01

1 Answers 1

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Note that, expanding the squares and bringing everything to the left side, you get

$$ x^2 - 8x + 16 + y^2 + 64 - 16y - 10 = 0 $$ Compare the standard form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ with what you get: $$ 1 x^2 + 0 xy + 1y^2 - 8x - 16y + 70 = 0 $$

You can write this as a quadratic equation $\mathbf{x}^T Q \mathbf{x} = 0$, where $$ Q = \left( \begin{array}{c c c} 1 & 0 & -4 \\ 0 & 1 & -8 \\ -4 & -8 & 70 \end{array} \right) $$ and $\mathbf{x} = \left( \begin{array}{c} x \\ y \\ 1 \end{array} \right) $.

The quadratic equation $\mathbf{x}^T Q \mathbf{x}$ corresponding to the standard form generally has $$ Q = \left( \begin{array}{c c c} A & B/2 & D/2 \\ B/2 & C & E/2 \\ D/2 & E/2 & F \end{array} \right) $$ Make sure you take a moment to write it down analytically and understand why you can write it as such.