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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