SATHEE: Straight Line and Pair of Straight Lines 1 Question 17

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 $△ P Q R$ is a line

(2019 Main, 10 Jan I)

(a) with slope $\frac{2}{3}$

(b) with slope $\frac{3}{2}$

(d) parallel to $X$ -axis

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Solution:

Let the coordinates of point $P$ be $(x_{1}, y_{1})$

$∵ P$ lies on the line $2 x - 3 y + 4 = 0$

$\begin{array}{ll} ∴ & 2 x_{1} - 3 y_{1} + 4 = 0 \\ \Rightarrow & y_{1} = \frac{2 x_{1} + 4}{3} \end{array}$

Now, let the centroid of $△ P Q R$ be $G (h, k)$ , then

$\begin{aligned} h & = \frac{x_{1} + 1 + 3}{3} \\ and & x_{1} & = 3 h - 4 \\ k & = \frac{y_{1} + 4 - 2}{3} \\ \Rightarrow & k & = \frac{\frac{2 x_{1} + 4}{3} + 2}{3} \\ \Rightarrow & 3 k & = \frac{2 x_{1} + 4 + 6}{3} \\ \Rightarrow & 9 k - 10 & = 2 x_{1} \end{aligned}$

[from Eq. (i)]

Now, from Eqs. (ii) and (iii), we get

$\begin{aligned} 2 (3 h - 4) & = 9 k - 10 \\ \Rightarrow & 6 h - 8 & = 9 k - 10 \\ \Rightarrow & 6 h - 9 k + 2 & = 0 \end{aligned}$

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 $\frac{2}{3} ∵$ slope of $a x + b y + c = 0$ is $- \frac{a}{b}$

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 $△ P Q R$ is a line

Solution:

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Straight Line and Pair of Straight Lines 1 Question 17

17. A point P moves on the line 2x−3y+4=0. If Q(1,4) and R(3,−2) are fixed points, then the locus of the centroid of △PQR is a line

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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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 $△ P Q R$ is a line