x^2+2xy+y^2=x
You asked:
Investigate the equation: \({x}^{2} + 2 x y + {y}^{2} = x\).
MathBot Answer:
The graph of the equation \({x}^{2} + 2 x y + {y}^{2} = x\) is a parabola.
The coordinates of its vertex are: \(\left(\frac{1}{16}, \frac{3}{16}\right)\)
The coordinates of its focus are: \(\left(\frac{1}{8}, \frac{1}{8}\right)\)
The directrix has equation: \(8 x - 8 y + 2 = 0\).
The parabola has parameterization: \[\begin{aligned}x\left(\theta\right) &= \frac{1}{16} + 2 \theta^{2} - \frac{\sqrt{2} \theta}{2}\\y\left(\theta\right) &= \frac{3}{16} - 2 \theta^{2} - \frac{\sqrt{2} \theta}{2}\end{aligned}\]