Find the equation of the lines joining the origin to the points of intersection of the straight line y=3x+2 with the curve $x^{2} + 2 x y + 3 y^{2} + 4 x + 8 y - 11 = 0$

$7 x^{2} - 2 x y + y^{2} = 0$

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$7 x^{2} - 2 x y - y^{2} = 0$

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$7 x^{2} + 2 x y + y^{2} = 0$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

$7 x^{2} + 2 x y + y^{2} = 0$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

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Solution

The correct option is B
$7 x^{2} - 2 x y - y^{2} = 0$

We will use method of homogenization to solve this. The combination of pairs straight lines is given by
$A x^{2} + 2 h x y + b y^{2} + 2 g x (\frac{l x + m y}{- n}) + 2 f y (\frac{l x + m y}{- n}) + c (\frac{l x + m y}{- n})^{2} = 0$

the term equivalent to $\frac{l x + m y}{- n}$ is $\frac{y - 3 x}{2}$ . We can take $\frac{(3 x - y}{- 2)}$ also.
$\Rightarrow x^{2} + 2 x y + 3 y^{2} + 4 x \frac{(y - 3 x}{2)} + 8 x \frac{(y - 3 x}{2)}$ -11x $\frac{(y - 3 x}{2)^{2}}$
$\Rightarrow x^{2} + 2 x y + 3 y^{2} + 2 x y - 6 x^{2} + 4 y^{2} - 12 x y - \frac{11}{4} (y^{2} + 9 x^{2} - 6 x y) = 0$
$\Rightarrow - 5 x^{2} + 7 y^{2} - 8 x y - \frac{11 y^{2} - 99 x^{2} + 66 x y}{4} = 0$
$\Rightarrow - 20 x^{2} + 28 y^{2} - 32 x y - 11 y^{2} - 99 x^{2} + 66 x y = 0$
$\Rightarrow - 119 x^{2} + 17 y^{2} + 34 x y = 0$
$\Rightarrow 7 x^{2} - 2 x y - y^{2} = 0$

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Find the equation of the lines joining the origin to the points of intersection of the straight line y=3x+2 with the curve x2+2xy+3y2+4x+8y−11=0

Find the equation of the lines joining the origin to the points of intersection of the straight line y=3x+2 with the curve $x^{2} + 2 x y + 3 y^{2} + 4 x + 8 y - 11 = 0$