Question

Find the ratio in which the point $P(2,y)$ divides the line segment joining the point $A(-2,2)$ and $B(3,7)$. Also find the value of $y$.

• A. $y=5$
• B. $4:1$
• C. $3:1$
• D. $y=6$

Answer 1

Answer: (B), (D)

The correct options are
B $4:1$
D $y=6$

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then $(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$
Let the ratio be $k:1$

Substituting $(x_1,y_1)=(-2,2)$ and $(x_2,y_2)=(3,7)$ in the section formula and equating its coordinate to point P, we get

$(\frac{k\left(3\right)+1(-2)}{k+1},\frac{k\left(7\right)+1\left(2\right)}{k+1})=(2,y)$

$(\frac{3k-2}{k+1},\frac{7k+2}{k+1})=(2,y)$

Comparing the $x$ - coordinate,
$\Rightarrow \frac{3k-2}{k+1}=2$
$\Rightarrow 3k-2=2k+2$

$\Rightarrow k=4$

Hence, the ratio is $4:1$.

Comparing the $y$ - coordinate,
$\Rightarrow \frac{7k+2}{k+1}=y$
$\Rightarrow \frac{7\left(4\right)+2}{4+1}=y$

$\Rightarrow y=6$