The length of the intercept on the x - axis cut by the pair of lines $2 x^{2} + 5 x y + 3 y^{2} + 6 x + 7 y + 1 = 0$ is

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Solution

$2 x^{2} + 5 x y + 3 y^{2} + 6 x + 7 y + 1 = 0$

$y = 0$

$2 x^{2} + 6 x + 1 = 0$

$\frac{- 6 \pm \sqrt{(36) - 4 (2) (1)}}{2 (2)}$

$\frac{- 6 \pm \sqrt{28}}{4} = \frac{- 3 \pm \sqrt{7}}{2}$

$x_{1} = \frac{- 3 - \sqrt{7}}{2}$ & $x_{2} = \frac{- 3 + \sqrt{7}}{2}$

$(x_{2} - x_{1}) = \frac{- 3 + \sqrt{7}}{2} - (\frac{- 3 - \sqrt{7}}{2})$

$= \frac{- 3}{2} + \frac{\sqrt{7}}{2} + \frac{3}{2} + \frac{\sqrt{7}}{2}$

$= \frac{2 \sqrt{7}}{2} = \sqrt{7}$ units.

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The length of the intercept on the x - axis cut by the pair of lines 2x2+5xy+3y2+6x+7y+1=0 is

2x2+5xy+3y2+6x+7y+1=0y=02x2+6x+1=0−6±√(36)−4(2)(1)2(2)−6±√284=−3±√72x1=−3−√72 & x2=−3+√72(x2−x1)=−3+√72−(−3−√72)=−32+√72+32+√72=2√72=√7 units.

The length of the intercept on the x - axis cut by the pair of lines $2 x^{2} + 5 x y + 3 y^{2} + 6 x + 7 y + 1 = 0$ is