If $a x^2+6 x y+b y^2-10 x+10 y-6=0$ represents a pair of perpendicular lines, then the values of $|a|$ equals

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Aicharya · Accepted Answer

Given pair of perpendicular lines are given as, $ a x^2+6 x y+b y^2-10 x+10 y-6=0 $ General equation is given as, $ a x^2+2 h x y+b y^2+2 g x+2 f y+c=0 $ Compare Eqs. (i) and (ii), we obtain $ \begin{aligned} & a=a, 2 h=6 \Rightarrow h=3, b=b, 2 g=-10 \Rightarrow g=-5 \ & 2 f=10 \Rightarrow f=5, c=-6 \ & \left|\begin{array}{lll} a & h & g \ h & b & f \ g & f & c \end{array}ight|=0 \Rightarrow\left|\begin{array}{ccc} a & 3 & -5 \ 3 & b & 5 \ -5 & 5 & -6 \end{array}ight|=0 \ & \Rightarrow a(-6 b-25)-3(-18+25)-5(15-5 b)=0 \ & \Rightarrow \quad 25 a+25 b+6 a b+96=0 \quad \ldots . \end{aligned} $ Since, lines are perpendicular $ \Rightarrow \quad a+b=0 	ext { or } a=-b $ Use Eq. (iv) in Eq. (iii), $ \begin{array}{rlrl} & 25(a-a)+6 a(-a)+96 & =0 \ \Rightarrow & & 6 a^2=96 \Rightarrow a^2 & =16 \ \Rightarrow & & & a= \pm 4 \ \Rightarrow & & & |a|=4 \end{array} $

If $a x^2+6 x y+b y^2-10 x+10 y-6=0$ represents a pair of perpendicular lines, then the values of $|a|$ equals

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