If the pair of straight lines given by A x^2+ | vedclass

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(A) The pair of straight lines $A x^2+2 H x y+B y^2=0$ passes through the origin. For these lines to form an equilateral triangle with the line $a x+b

Vedclass · Accepted Answer

$4$

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(A) The pair of straight lines $A x^2+2 H x y+B y^2=0$ passes through the origin. For these lines to form an equilateral triangle with the line $a x+b y+c=0$,the angle between the pair of lines must be $60^{\circ}$ or $\frac{\pi}{3}$.The angle $	heta$ between the lines $A x^2+2 H x y+B y^2=0$ is given by $	an 	heta = \frac{2 \sqrt{H^2-A B}}{|A+B|}$.Setting $	heta = \frac{\pi}{3}$,we have $	an \frac{\pi}{3} = \sqrt{3}$.So,$\sqrt{3} = \frac{2 \sqrt{H^2-A B}}{|A+B|}$.Squaring both sides,we get $3 = \frac{4(H^2-A B)}{(A+B)^2}$.$3(A+B)^2 = 4(H^2-A B)$.$3(A^2+2 A B+B^2) = 4 H^2-4 A B$.$3 A^2+6 A B+3 B^2 = 4 H^2-4 A B$.$3 A^2+10 A B+3 B^2 = 4 H^2$.Factoring the left side: $3 A^2+9 A B+A B+3 B^2 = 4 H^2$.$3 A(A+3 B)+B(A+3 B) = 4 H^2$.$(A+3 B)(3 A+B) = 4 H^2$.

If the pair of straight lines given by $A x^2+2 H x y+B y^2=0$,where $(H^2 > A B)$,forms an equilateral triangle with the line $a x+b y+c=0$,then $(A+3 B)(3 A+B)=$ (in $H^2$)

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