Answer
(a) ellipse.
(b) $\phi\approx53.13^{\circ}$.
$X^2+\frac{(Y+1)^2}{4}=1$
(c) See graph.
Work Step by Step
(a) Rewrite the original equation as $52x^2+72xy+73y^2-40x+30y-75=0$ and we have $A=52, B=72, C=73, D=-40,E=-30,F=750$. The discriminant is $B^2-4AC=72^2-4\times52\times73=-10000\lt0$, thus the graph of the equation is an ellipse.
(b) To eliminate the xy-term, we need a rotation of axes with an angle $\phi$ where $cot2\phi=\frac{A-C}{B}=\frac{52-73}{72}=-\frac{7}{24}$ which gives $2\phi\approx106.26^{\circ}$ and $\phi\approx53.13^{\circ}$.
With $tan2\phi=-\frac{24}{7}$, we have $cos2\phi=-\frac{7}{25}$ and $sin\phi=\sqrt {\frac{1-cos2\phi}{2}}=\frac{4}{5}$ and $cos\phi=\sqrt {\frac{1+cos2\phi}{2}}=\frac{3}{5}$
The transformation formula gives:
$x=X\cdot \frac{3}{5} - Y\cdot \frac{4}{5}=\frac{3}{5}X-\frac{4}{5}Y$,
$y=X\cdot \frac{4}{5}+Y\cdot\frac{3}{5}=\frac{4}{5}X+\frac{3}{5}Y$
Use them in the original equation to get $52(\frac{3}{5}X-\frac{4}{5}Y)^2+72(\frac{3}{5}X-\frac{4}{5}Y)(\frac{4}{5}X+\frac{3}{5}Y)+73(\frac{4}{5}X+\frac{3}{5}Y)^2-40(\frac{3}{5}X-\frac{4}{5}Y)-30(\frac{4}{5}X+\frac{3}{5}Y)+75=0$
Simplify this equation by multiplying 25 and combine like terms to get $2500X^2+625Y^2+1250Y-1875=0$ or $4X^2+Y^2+2Y-3=0$ which can be rewrite as $X^2+\frac{(Y+1)^2}{4}=1$
(c) See graph.
