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  • If the equation <img alt="\mathrm{ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bax%5E%7B2%7D%20&plus;%202hxy%20&plus;%20by%5E

If the equation \mathrm{ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c = 0} represents a pair of parallel lines, then the distance between them is

 

Option: 1

\mathrm{\sqrt{\frac{g^2-a c}{h^2+a^2}}}
 


Option: 2

\mathrm{\sqrt[2]{\frac{g^2-a c}{h^2+a^2}}}

 


Option: 3

\mathrm{\sqrt{\frac{g^2+a c}{a(a+b)}}}
 


Option: 4

\mathrm{\sqrt[2]{\frac{\mathrm{g}^2+a c}{a(a+b)}}}


Answers (1)

best_answer

Since, \mathrm{a x^2+2 h x y+b y^2+2 g x+2 f y+c^2=0}   .......(i)

represents pair of lines, therefore, \mathrm{h^2=a b}

Let \mathrm{\mathrm{ax} 2+2 \mathrm{hxy}+\mathrm{by} 2+2 \mathrm{gx}+2 \mathrm{fy}+\mathrm{c} \equiv\left(\sqrt{a} x+\sqrt{b} y+c_1\right)\left(\sqrt{a} x+\sqrt{b} y+c_2\right)}
On comparing the coefficients of \mathrm{\mathrm{x}} and constant terms, we get

\mathrm{ \sqrt{a}\left(c_1+c_2\right)=2 g }

\mathrm{ c_1+c_2=\frac{2 g}{\sqrt{a}} \text { and } c 1 c 2=c }
Now, distance between the parallel lines is

\mathrm{ \frac{\left|c_1-c_2\right|}{\sqrt{(\sqrt{a})^2+(\sqrt{b})^2}}=\frac{\sqrt{\frac{4 g^2}{a}-4 c}}{\sqrt{a+b}}=2 \sqrt{\frac{g^2-a c}{a(a+b)}}=2 \sqrt{\frac{g^2-a c}{a^2+h^2}} }

\mathrm{ (h^2=a b ) }

Hence option 2 is correct.

 

Posted by

Nehul

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