# Current Affairs JEE Main & Advanced

## Equation of Pair of Straight Lines

Category : JEE Main & Advanced

(1) Equation of a pair of straight lines passing through origin : The equation $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$ represents a pair of straight line passing through the origin where $a,\,\,h,\,\,b$ are constants.

Let the lines represented by $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$ be $y-{{m}_{1}}x=0,\,\,y-{{m}_{2}}x=0$. Then, ${{m}_{1}}+{{m}_{2}}=-\frac{2h}{b}$ and ${{m}_{1}}{{m}_{2}}=\frac{a}{b}$

Then, two straight lines represented by $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$ are $ax+hy+y\sqrt{{{h}^{2}}-ab}=0$ and $ax+hy-y\sqrt{{{h}^{2}}-ab}=0$.

Hence, (a) The lines are real and distinct, if ${{h}^{2}}-ab>0$

(b) The lines are real and coincident, if ${{h}^{2}}-ab=0$

(c) The lines are imaginary, if ${{h}^{2}}-ab<0$

(2) General equation of a pair of straight lines : An equation of the form, $a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0$ where $a,\,\,b,\,\,c,\,\,f,\,\,g,\,\,h$ are constants, is said to be a general equation of second degree in $x$ and $y$.

The necessary and sufficient condition for $a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0$ to represents a pair of straight lines is that $abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0$ or $\left| \begin{matrix} a & h & g \\ h & b & f \\ g & f & c \\ \end{matrix} \right|=0$.

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