Question

Column II gives the coordinates of the point P that divides the line segment joining the points given in column I, match them correctly

Answer 1

We shall apply the section formula

$P(x,y)=(\frac{n{x}_1+m{x}_2}{m+n},\frac{n{y}_1+m{y}_2}{m+n}.)$

When the point $P(x,y)$ divides a line segment AB joining $A(x_1,y_1)\&B(x_2,y_2)$ internally in the ratio $m:n$

A) Let the line segment AB joining $A(x_1,y_1)=(-1,3)$ & $B(x_2,y_2)=(5,-6)$ internally in the ratio $m:n=1:2$.

Then, by the section formula,

$x=\frac{n{x}_1+m{x}_2}{m+n}=\frac{2\times (-1)+1\times \left(5\right)}{1+2}=1$ and $y=\frac{n{y}_1+m{y}_2}{m+n}=\frac{2\times 3+1\times (-6)}{1+2}=0$

So $P(x,y)=(1,0)$, which matches with 4.

B) Let the line segment AB joining $A(x_1,y_1)=(-2,1)$ & $B(x_2,y_2)=(1,4)$ internally in the ratio $m:n=2:1$.

Then, by the section formula,

$x=\frac{n{x}_1+m{x}_2}{m+n}=\frac{1\times (-2)+2\times \left(1\right)}{1+2}=0$ and $y=\frac{n{y}_1+m{y}_2}{m+n}=\frac{1\times 1+2\times \left(4\right)}{1+2}=3$

So, $P(x,y)=(0,3)$ which matches with 2.

C) Let the line segment AB joining $A(x_1,y_1)=(-1,7)$ & $B(x_2,y_2)=(4,-3)$ internally in the ratio $m:n=2:3$

Then, by the section formula,

$x=\frac{n{x}_1+m{x}_2}{m+n}=\frac{3\times (-1)+2\times 4}{2+3}=1$ and $y=\frac{n{y}_1+m{y}_2}{m+n}=\frac{3\times 7+2\times (-3)}{2+3}=3$

So, $P(x,y)=(1,3)$ which matches with 3.

D) Let the line segment AB joining $A(x_1,y_1)=(4,-3)$ & $B(x_2,y_2)=(8,5)$ internally in the ratio $m:n=3:1$

Then, by the section formula,

$x=\frac{n{x}_1+m{x}_2}{m+n}=\frac{1\times \left(4\right)+3\times \left(8\right)}{3+1}=7$ and $y=\frac{n{y}_1+m{y}_2}{m+n}=\frac{1\times (-3)+3\times 5}{3+1}=3$

So, $P(x,y)=(7,3)$ which matches with 1.

$A⟶$ matches with $4$

$B⟶$ matches with $2$

$C⟶$ matches with $3$

$D⟶$ matches with $1$