Answer
(a) parabola.
(b) $\phi\approx73.74^{\circ}$.
$Y=-X^2+\frac{1}{25}$
(c) See graph.
Work Step by Step
(a) Rewrite the original equation as $49x^2+336xy+576y^2-600x+175y-25=0$ and we have $A=49, B=336, C=576, D=-600,E=175,F=-25$. The discriminant is $B^2-4AC=336^2-4\times49\times576=0$, thus the graph of the equation is a parabola.
(b) To eliminate the xy-term, we need a rotation of axes with an angle $\phi$ where $cot2\phi=\frac{A-C}{B}=\frac{49-576}{336}=-\frac{527}{336}$ which gives $2\phi\approx147.48^{\circ}$ and $\phi\approx73.74^{\circ}$.
With $tan2\phi=-\frac{336}{527}$, we have $cos2\phi=-\frac{527}{625}$ and $sin\phi=\sqrt {\frac{1-cos2\phi}{2}}=\frac{24}{25}$ and $cos\phi=\sqrt {\frac{1+cos2\phi}{2}}=\frac{7}{25}$
The transformation formula gives:
$x=X\cdot \frac{7}{25} - Y\cdot \frac{24}{25}=\frac{1}{25}(7X-24Y)$,
$y=X\cdot \frac{24}{25}+Y\cdot \frac{7}{25}=\frac{1}{25}(24X+7Y)$
Use them in the original equation to get $49(\frac{1}{25}(7X-24Y))^2+336(\frac{1}{25}(7X-24Y))(\frac{1}{25}(24X+7Y))+576(\frac{1}{25}(24X+7Y))^2-600(\frac{1}{25}(7X-24Y))+175(\frac{1}{25}(24X+7Y))-25=0$
Simplify this equation by multiplying 625 and combine like terms to get $390625X^2+390625Y-25\times625=0$
or $Y=-X^2+\frac{1}{25}$
(c) See graph.
