Straight Line and Pair of Straight Lines 1 Question 17
17. A point $P$ moves on the line $2 x-3 y+4=0$. If $Q(1,4)$ and $R(3,-2)$ are fixed points, then the locus of the centroid of $\triangle P Q R$ is a line
(2019 Main, 10 Jan I)
(a) with slope $\frac{2}{3}$
(b) with slope $\frac{3}{2}$
(c) parallel to $Y$-axis
(d) parallel to $X$-axis
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Solution:
- Let the coordinates of point $P$ be $\left(x _1, y _1\right)$
$\because P$ lies on the line $2 x-3 y+4=0$
$$ \begin{array}{ll} \therefore & 2 x _1-3 y _1+4=0 \\ \Rightarrow & y _1=\frac{2 x _1+4}{3} \end{array} $$
Now, let the centroid of $\triangle P Q R$ be $G(h, k)$, then
$$ \begin{array}{rlrl} & & h & =\frac{x _1+1+3}{3} \\ & \text { and } & x _1 & =3 h-4 \\ & k & =\frac{y _1+4-2}{3} \\ \Rightarrow & k & =\frac{\frac{2 x _1+4}{3}+2}{3} \\ \Rightarrow & 3 k & =\frac{2 x _1+4+6}{3} \\ \Rightarrow & & 9 k-10 & =2 x _1 \end{array} $$
[from Eq. (i)]
Now, from Eqs. (ii) and (iii), we get
$$ \begin{array}{rlrl} & & 2(3 h-4) & =9 k-10 \\ \Rightarrow & & 6 h-8 & =9 k-10 \\ \Rightarrow & 6 h-9 k+2 & =0 \end{array} $$
Now, replace $h$ by $x$ and $k$ by $y$.
$\Rightarrow 6 x-9 y+2=0$, which is the required locus and slope of this line is $\frac{2}{3} \quad \because$ slope of $a x+b y+c=0$ is $-\frac{a}{b}$
