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  • Equation represents a pair of lines. Combined equation of lines t

Equation \mathrm{ax^ 2+2 hxy+ by ^2=0} represents a pair of lines. Combined equation of lines that can be obtained by reflecting these lines about the x-axis is -
 

Option: 1

\mathrm{\mathrm{ax} 2-2 \mathrm{hxy}+by 2=0}



 


Option: 2

\mathrm{bx} 2+2 \mathrm{hxy}+\mathrm{ay} 2=0 \mathrm{~b}


Option: 3

\mathrm{bx 2-2 h x y+a y 2=0}


Option: 4

\text{ None of these}


Answers (1)

best_answer

Let the lines represented by \mathrm{a x^2+2 h x y+b y^2=0 \: be\: y=m 1 x \, and\: y=m_2 x} then \mathrm{m_1+m 2=-\frac{2 h}{b}, \mathrm{m} 1 \mathrm{~m} 2=\frac{a}{b}} If these lines are reflected about the x-axis, their equation becomes \mathrm{y+m_1 x=0, y+m 2 x=0} and their combined equation is

\mathrm{(\mathrm{y}+\mathrm{m} 1 \mathrm{x})(\mathrm{y}+\mathrm{m} 2 \mathrm{x})=0}

\mathrm{ \Rightarrow \mathrm{y}^2+\mathrm{xy}(\mathrm{m} 1+\mathrm{m} 2)+\mathrm{m} 1 \mathrm{~m} 2 \mathrm{x} 2=0 }

\mathrm{ \text { i.e., } b y^2-2 h x y+a x^2=0 }

Hence option 4 is correct.

 

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