Find the angle between the lines joining the origin to the points of intersection of the curve x² + 2xy + y² + 2x + 2y – 5 = 0 and the line 3x – y + 1 = 0.

The given curve is x²+2xy + y² + 2x + 2y–5 = 0
And the given line is 3x – y + 1 = 0

$\Rightarrow 3\mathrm{x}-\mathrm{y}=-1\Rightarrow \mathrm{y}-3\mathrm{x}=1 ...\left(2\right)$
The given curve (1) and the line (2) intersects at A, B and join $\overline{\mathrm{OA}}\text{ and }\overline{\mathrm{OB}}$
By homogenizing the curve (1) with the line (2) we get
$\begin{array}{l}{\mathrm{x}}^2+2\mathrm{xy}+{\mathrm{y}}^2+2\mathrm{x}\left(1\right)+2\mathrm{y}\left(1\right)-5(1{)}^2=0\\ \Rightarrow {\mathrm{x}}^2+2\mathrm{xy}+{\mathrm{y}}^2+2\mathrm{x}(\mathrm{y}-3\mathrm{x})+2\mathrm{y}(\mathrm{y}-3\mathrm{x})-5(\mathrm{y}-3\mathrm{x}{)}^2=0\\ \\ \Rightarrow {\mathrm{x}}^2+2\mathrm{xy}+{\mathrm{y}}^2+2\mathrm{xy}-6{\mathrm{x}}^2+2{\mathrm{y}}^2-6\mathrm{xy}-5\left({\mathrm{y}}^2-6\mathrm{xy}+9{\mathrm{x}}^2\right)=0\\ \\ \Rightarrow {\mathrm{x}}^2+2\mathrm{xy}+{\mathrm{y}}^2+2\mathrm{xy}-6{\mathrm{x}}^2+2{\mathrm{y}}^2-6\mathrm{xy}-5y^2+30xy-45x^2=0\\ \Rightarrow -50{\mathrm{x}}^2+28\mathrm{xy}-2{\mathrm{y}}^2=0\\ \Rightarrow 25{\mathrm{x}}^2-14\mathrm{xy}+{\mathrm{y}}^2=0\dots .\left(3\right)\end{array}$
Equation (3) represents the pair of lines $\overline{\mathrm{OA}},\overline{\mathrm{OB}}$
Comparing equation (3) with
${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2=0$;
$\text{ we get }\mathrm{a}=25;2\mathrm{h}=-14;\mathrm{b}=1$
let $\mathrm{\theta }$ be the angle between the pair of lines (3)
$\begin{array}{l}\therefore \cos \mathrm{\theta }=\frac{|\mathrm{a}+\mathrm{b}|}{\sqrt{(\mathrm{a}-\mathrm{b}{)}^2+4{\mathrm{h}}^2}}\\ \cos \mathrm{\theta }=\frac{|25+1|}{\sqrt{(25-1{)}^2+4(7{)}^2}}[\because 2\mathrm{h}=-14\Rightarrow \mathrm{h}=-7]\\ =\frac{26}{\sqrt{576+4\times 49}}=\frac{\left|26\right|}{\sqrt{576+196}}\\ =\frac{\left|26\right|}{\sqrt{772}}=\frac{\left|26\right|}{\sqrt{4\times 193}}=\frac{13}{\sqrt{193}}\\ \therefore \mathrm{\theta }={\cos }^{-1}\left(\frac{13}{\sqrt{193}}\right)\end{array}$