Question

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 (5/3, q) respectively then the value of p and q is

• A. $p=7/3,q=-3$
• B. $p=7/3,q=0$
• C. $p=5/3,q=0$
• D. None of these

Answer 1

The correct option is B $p=7/3,q=0$

Suppose Points P and Q trisect the line segment joining the points$A(3,-4)$ and $B(1,2)$
This means, P divides AB in the ratio $1:2$ and Q divides it in the ratio $2:1$

Using the section formula, if point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ internally in the ratio $m:n$, then$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Substituting $(x_1,y_1)=(3,-4)$ and $(x_2,y_2)=(1,2)$ and $m=1,n=2$ in the section formula, we get the point P $=(\frac{1\left(1\right)+2\left(3\right)}{1+2},\frac{1\left(2\right)+2(-4)}{1+2})=(\frac{7}{3},-2)$

Given. $P(p,-2)=(\frac{7}{3},-2)$

$=>p=\frac{7}{3}$

Substituting $(x_1,y_1)=(3,-4)$ and $(x_2,y_2)=(1,2)$ and $m=2,n=1$ in the section formula, we get the point Q $=(\frac{2\left(1\right)+1\left(3\right)}{2+1},\frac{2\left(2\right)+1(-4)}{2+1})=(\frac{5}{3},0)$

Given. $Q(\frac{5}{3},0)=(\frac{5}{3},q)$

$=>q=0$