Find the real points $(x, y)$ satisfying $3 x^{2} + 3 y^{2} - 4 x y + 10 x - 10 y + 10 = 0$

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Solution

Given $3 x^{2} + 3 y^{2} - 4 x y + 10 x - 10 y + 10 = 0$ represents pair of straight lines
The point of intersection for $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ is $(\frac{h f - b g}{a b - h^{2}}, \frac{g h - a f}{a b - h^{2}})$
POI is $(\frac{(- 2) (- 5) - (3) (5)}{(3) (3) - (- 2)^{2}}, \frac{(5) (- 2) - (3) (- 5)}{(3) (3) - (- 2)^{2}}) = (- 1, 1)$

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Find the real points (x,y) satisfying 3x2+3y2−4xy+10x−10y+10=0

Given 3x2+3y2−4xy+10x−10y+10=0 represents pair of straight lines The point of intersection for ax2+2hxy+by2+2gx+2fy+c=0 is (hf−bgab−h2,gh−afab−h2)POI is ((−2)(−5)−(3)(5)(3)(3)−(−2)2,(5)(−2)−(3)(−5)(3)(3)−(−2)2)=(−1,1)

Find the real points $(x, y)$ satisfying $3 x^{2} + 3 y^{2} - 4 x y + 10 x - 10 y + 10 = 0$