and 
is trisected at the points 
and ![\[Q\left(\frac{5}{3}, q\right) \text {. Find the values of } p \text { and } q \text {. }\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-53b985e3570bde212d9bb8fae5d7a8a2_l3.png "Rendered by QuickLaTeX.com")
Let 
and 
be the points of trisection of 
.
=> P divides in the radio 
So, the coordinates of are
![\[\begin{aligned} &x=\frac{\left(m x_{2}+n x_{1}\right)}{(m+n)}, y=\frac{\left(m y_{2}+n y_{1}\right)}{(m+n)} \\ &\Rightarrow x=\frac{\{1 \times 1+2 \times(3)\}}{1+2}, y=\frac{\{1 \times 2+2 \times(-4)\}}{1+2} \\ &\Rightarrow x=\frac{1+6}{3}, y=\frac{2-8}{3} \\ &\Rightarrow x=\frac{7}{3}, y=-\frac{6}{3} \\ &\Rightarrow x=\frac{7}{3}, y=-2 \end{aligned}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-7f5c1ae67cb1b8134ab577331c13675e_l3.png "Rendered by QuickLaTeX.com")
=> the coordinates of 
are 
are the coordinates of .
Q divides the line AB in the ratio 
=> the coordinates of are
![\[\begin{aligned} &\mathrm{x}=\frac{\left(\mathrm{mx}_{2}+\mathrm{mx}_{1}\right)}{(\mathrm{m}+\mathrm{n})}, \mathrm{y}=\frac{\left(\mathrm{my}_{2}+\mathrm{my}_{1}\right)}{(\mathrm{m}+\mathrm{n})} \\ &\Rightarrow \mathrm{x}=\frac{(2 \times 1+1 \times 3)}{2+1}, \mathrm{y}=\frac{\{2 \times 2+1 \times(-4)\}}{2+1} \\ &\Rightarrow \mathrm{x}=\frac{2+3}{3}, \mathrm{y}=\frac{4-4}{3} \\ &\Rightarrow \mathrm{x}=\frac{5}{3}, \mathrm{y}=0 \end{aligned}\]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-f38b67b325a723d2cf61ef2dfae85944_l3.png "Rendered by QuickLaTeX.com")
=> coordinates of are 
.
But the given coordinates of are 
Therefore, and 