The product of the perpendicular distances from the origin on the pair of straight lines $12 x^2+25 x y+12 y^2+10 x+11 y+2=0$, is

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Aicharya · Accepted Answer

Given that

First we take homogeneous part of Eq. (i), $\begin{aligned} & \text { i.e. } 12 x^2+25 x y+12 y^2=0 \\ & \Rightarrow(3 x+4 y)(4 x+3 y)=0 \end{aligned}$ So, let the lines represented by Eq. (i) be

The combined Eqs. of (ii) and (iii), we get $\begin{gathered} \left(3 x+4 y+c_1\right)\left(4 x+3 y+c_2\right)=0 \\ \Rightarrow \quad(3 x+4 y)(4 x+3 y)+c_1(4 x+3 y)+ \\ c_2(3 x+4 y)+c_1 c_2=0 \\ \Rightarrow \quad 12 x^2+25 x y+12 y^2+\left(4 c_1+3 c_2\right) x \\ +\left(3 c_1+4 c_2\right) y+c_1 c_2=0 \end{gathered}$ On comparing the equation with Eq. (i), we get

The perpendicular distance from origin to the equations (vi) and (vii) are $\begin{aligned} & p_1=\frac{|0+0+1|}{\sqrt{3^2+4^2}}=\frac{1}{5} \text { and } p_2=\frac{|0+0+2|}{\sqrt{4^2+3^2}}=\frac{2}{5} \\ \therefore \quad & p_1 \cdot p_2=\frac{1}{5} \cdot \frac{2}{5}=\frac{2}{25} \end{aligned}$

The product of the perpendicular distances from the origin on the pair of straight lines $12 x^2+25 x y+12…

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