Question
Find the ratio in which the point (2, y) divides the line segment joining the points A(-2, 2) and B(3, 7). Also, find the value of y.
✓
Answer
Let the point P(2, y) divide the line segment joining the points A(-2, 2) and B(3, 7) in the ratio k : 1
Then, the coordinates of P are,
$\bigg[\frac{3\text{k}+(-2)\times1}{\text{k+1}},\frac{7\text{k}+2\times1}{\text{k+1}}\bigg]$
$\bigg[\frac{3\text{k}-2}{\text{k+1}},\frac{7\text{k}+2}{\text{k}+1}\bigg]$
But the coordinates of P are given as (2, y).
$\therefore\ \frac{3\text{k}-2}{\text{k}+1}=2$
$\Rightarrow\ 3\text{k}-2=2\text{k}+2$
$\Rightarrow\ 3\text{k}-2\text{k}=2+2$
$\Rightarrow\ \text{k}=4$
$\frac{7\text{k}+2}{\text{k}+1}=\text{y}$
Putting the value of k, we get
$\frac{7\times4+2}{4+1}=\text{y}$
$\frac{30}{5}=\text{y}$
$6=\text{y}$
i.e., $\text{y}=6$
Hence, the ratio is 4 : 1 and y = 6.
Then, the coordinates of P are,
$\bigg[\frac{3\text{k}+(-2)\times1}{\text{k+1}},\frac{7\text{k}+2\times1}{\text{k+1}}\bigg]$
$\bigg[\frac{3\text{k}-2}{\text{k+1}},\frac{7\text{k}+2}{\text{k}+1}\bigg]$
But the coordinates of P are given as (2, y).
$\therefore\ \frac{3\text{k}-2}{\text{k}+1}=2$
$\Rightarrow\ 3\text{k}-2=2\text{k}+2$
$\Rightarrow\ 3\text{k}-2\text{k}=2+2$
$\Rightarrow\ \text{k}=4$
$\frac{7\text{k}+2}{\text{k}+1}=\text{y}$
Putting the value of k, we get
$\frac{7\times4+2}{4+1}=\text{y}$
$\frac{30}{5}=\text{y}$
$6=\text{y}$
i.e., $\text{y}=6$
Hence, the ratio is 4 : 1 and y = 6.
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