- Trending Categories
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
The line segment joining the points $(3, -4)$ and $(1, 2)$ is trisected at the points $P$ and $Q$. If the coordinates of $P$ and $Q$ are $(p, -2)$ and $(\frac{5}{3}, q)$ respectively. Find the values of $p$ and $q$.
Given:
The line segment joining the points $(3, -4)$ and $(1, 2)$ is trisected at the points $P$ and $Q$. The coordinates of $P$ and $Q$ are $(p, -2)$ and $(\frac{5}{3}, q)$ respectively.
To do:
We have to find the values of $p$ and $q$.
Solution:
Let \( \mathrm{AB} \) be the line segment whose ends points are \( \mathrm{A}(3,-4) \) and \( \mathrm{B}(1,2) \)
The coordinates of \( \mathrm{P} \) and \( \mathrm{Q} \) which trisect \( \mathrm{AB} \) are \( \mathrm{P}(p,-2) \) and \( \mathrm{Q}\left(\frac{5}{3}, q\right) \).
Therefore,
$P$ divides \( A B \) in the ratio $1: 2$.
Using section formula, we get,
\( (x,y)=(\frac{m x_{2}+n x_{1}}{m+n}, \frac{m y_{2}+n y_{1}}{m+n}) \)
The coordinates of \( \mathrm{P} \) are \( (\frac{1 \times 1+2 \times 3}{1+2}, \frac{1 \times 2+2 \times (-4)}{1+2} \)
\( (p,-2)=(\frac{1+6}{3}, \frac{2-8}{3}) \)
\( =(\frac{7}{3}, -2) \)
This implies, \( p=\frac{7}{3} \)
Similarly,
$Q$ divides \( \mathrm{AB} \) in the ratio $2: 1$.
The coordinates of \( \mathrm{P} \) are \( (\frac{2 \times 1+1 \times 3}{1+2}, \frac{2 \times 2+1 \times (-4)}{1+2} \)
\( (\frac{5}{3},q)=(\frac{2+3}{3},\frac{4-4}{3} \)
\( =(\frac{5}{3},\frac{0}{3} \)
This implies,
\( q=0 \)
The values of $p$ and $q$ are $\frac{7}{3}$ and $0$ respectively.
- Related Articles
- Let P and Q be the points of trisection of the line segment joining the points $A( 2,\ -2)$ and $B( -7,\ 4)$ such that P is nearer to A. Find the coordinates of P and Q.
- Points $P$ and $Q$ trisect the line segment joining the points $A(-2,0)$ and $B(0,8)$ such that $P$ is near to $A$. Find the coordinates of $P$ and $Q$.
- Points $P, Q, R$ and $S$ divide the line segment joining the points $A (1, 2)$ and $B (6, 7)$ in 5 equal parts. Find the coordinates of the points $P, Q$ and $R$.
- Subtract \( 4 p^{2} q-3 p q+5 p q^{2}-8 p+7 q-10 \) from \( 18-3 p-11 q+5 p q-2 p q^{2}+5 p^{2} q \).
- Given that \( \frac{4 p+9 q}{p}=\frac{5 q}{p-q} \) and \( p \) and \( q \) are both positive. The value of $\frac{p}{q}$ is
- If $p=-2,\ q=-1$ and $r=3$, find the value of $p-q-r$.
- A line intersects the y-axis and x-axis at the points P and Q respectively. If $( 2,\ -5)$ is the mid-point then find the coordinates of P and Q.
- If $p=-2,\ q=-1$ and $r=3$, find the value of $p^{2}+q^{2}-r^{2}$.
- If p, q are real and p≠q, then show that the roots of the equation $(p-q)x^2+5(p+q)x-2(p-q)=0$ are real and unequal.
- If $p=-2,\ q=-1$ and $r=3$, find the value of $2 p^{2}-q^{2}+3 r^{2}$.
- The line joining the points $(2, 1)$ and $(5, -8)$ is trisected at the points P and Q. If point P lies on the line $2x – y + k = 0$. Find the value of $k$.
- Solve \( 2 p^{2} q^{2}-3 p q+4,5+7 p q-3 p^{2} q^{2} \).
- Factorise \( 16(2 p-3 q)^{2}-4(2 p-3 q) \).
- In the figure, \( O Q: P Q=3: 4 \) and perimeter of \( \Delta P O Q=60 \mathrm{~cm} \). Determine \( P Q, Q R \) and \( O P \)."\n
- A line intersects the \( y \)-axis and \( x \)-axis at the points \( \mathrm{P} \) and \( \mathrm{Q} \), respectively. If \( (2,-5) \) is the mid-point of \( P Q \), then the coordinates of \( P \) and \( Q \) are, respectively(A) \( (0,-5) \) and \( (2,0) \)(B) \( (0,10) \) and \( (-4,0) \)(C) \( (0,4) \) and \( (-10,0) \)(D) \( (0,-10) \) and \( (4,0) \)
