If a straight line drawn through the point of intersection of the lines 4 x+3 y-1=0 and 3 …

Question

If a straight line drawn through the point of intersection of the lines $4 x+3 y-1=0$ and $3 x+4 y-1=0$, meets the co-ordinate axes at the points P and Q , then the locus of the mid point of PQ is :

Accepted Answer

1. A straight line $L$ passes through the point of intersection of 2 straight lines: $ \begin{gathered} {L}_{1}:4x+3y-1=0 \ {L}_{2}:3x+4y-1=0 \end{gathered}$

2. The straight line $L$ cuts the axes at $P$ and $Q$

3. To identify the locus of the mid-point of the segment $PQ$

The point of intersection of ${L}_{1}$ and ${L}_{2}$ is $I(\alpha ,\beta )$

The mid-point of the segment $PQ$ is $M(h,k)$

Analytical Study :

We first solve the pair of straight lines ${L}_{1}$ and ${L}_{2}$ to locate the point of intersection :

$\begin{gathered} 4x+3y-1=0\Rightarrow 12x+9y=3 \ 3x+4y-1=0\Rightarrow 12x+16y=4 \end{gathered}\, \Rightarrow \beta =\frac{1}{7}\Rightarrow \alpha =\frac{1}{7}\Rightarrow I(\frac{1}{7},\frac{1}{7})\ldots ..(i)$

The straight line $L$ passes through the point $I$ identified by equation (i)

The equation of the straight line $L$ in point slope form is written below :

$L:(y-\frac{1}{7})=m(x-\frac{1}{7})\equiv L:7mx-7y=m-1\equiv L:\frac{7mx}{m-1}+\frac{7y}{1-m}=1\ldots .(ii)$

The equation (ii) gives the straight line $L$ in intercept form where the $X$ and $Y$ intercepts are observed below :

$\begin{gathered} P Q \ (\frac{m-1}{7m},0) (0,\frac{1-m}{7})1 \end{gathered}$
From Chart 1, the mid-point of the section $PQ$ is derived applying mid-point formulae :

$M(h,k)=M(\frac{m-1}{14m},\frac{1-m}{14})\equiv \begin{gathered} h \quad  k \ M(h,k) \quad  \frac{m-1}{14m} \quad  \frac{1-m}{14} \end{gathered}\, \ldots .2$
Chart 2 has the $X$ and $Y$ coordinate each as a function of $m$ where we can eliminate $m$ and mathematically trace the locus :

$k=\frac{1-m}{14}\Rightarrow m=1-14k\Rightarrow h=\frac{1-14k-1}{14(1-14k)}=\frac{14k}{14(14k-1)}=\frac{k}{(14k-1)}\Rightarrow 14hk-h=k\Rightarrow h+k-14hk=0$

The locus of the mid-point of the section $PQ$ after replacement of the dummy variables is found to be the rectangular hyperbola $x+y-14xy=0$

Solutions · 1