Question

The ratio in which the line segment joining the points $\left( { - 3,10} \right)$ and $\left( {6, - 8} \right)$ is divided by the point $\left( { - 1,6} \right)$ is $3:7$.
(a) True
(b) False
(c) Ambiguous
(d) Data insufficient

Answer 1

Here, we will use the section formula to find the ratio in which the point $\left( { - 1,6} \right)$ divide the line segment joining the points $\left( { - 3,10} \right)$ and $\left( {6, - 8} \right)$. Then we will check if the given statement is true, false or ambiguous.

Formula Used:
According to the section formula, the co-ordinates of a point dividing the line segment joining two points $P\left( {{x_1},{y_1}} \right)$ and $Q\left( {{x_2},{y_2}} \right)$ in the ratio $m:n$, are given by $\left( {\dfrac{{m{x_1} + n{x_2}}}{{m + n}},\dfrac{{m{y_1} + n{y_2}}}{{m + n}}} \right)$.

Complete step-by-step answer:
Let the point $\left( { - 1,6} \right)$ divide the line segment joining the points $\left( { - 3,10} \right)$ and $\left( {6, - 8} \right)$ in the ratio $k:1$.
The co-ordinates of the point dividing the line segment joining the points $\left( { - 3,10} \right)$ and $\left( {6, - 8} \right)$ in the ratio $k:1$, are $\left( { - 1,6} \right)$.
Comparing the point $\left( { - 3,10} \right)$ and $\left( {{x_1},{y_1}} \right)$, we get
${x_1} = - 3$ and ${y_1} = 10$
Comparing the point $\left( {6, - 8} \right)$ and $\left( {{x_2},{y_2}} \right)$, we get
${x_2} = 6$ and ${y_2} = - 8$
Comparing the ratios $k:1$ and $m:n$, we get
$m = k$ and $n = 1$
Therefore, substituting ${x_1} = - 3$, ${y_1} = 10$, ${x_2} = 6$, ${y_2} = - 8$, $m = k$ and $n = 1$ in the section formula, $\left( {\dfrac{{m{x_1} + n{x_2}}}{{m + n}},\dfrac{{m{y_1} + n{y_2}}}{{m + n}}} \right)$, we get
$\Rightarrow \left( { - 1,6} \right) = \left( {\dfrac{{k\left( { - 3} \right) + 1\left( 6 \right)}}{{k + 1}},\dfrac{{k\left( {10} \right) + 1\left( { - 8} \right)}}{{k + 1}}} \right)$
Comparing the abscissa and the ordinate, we get the equations
$\Rightarrow - 1 = \dfrac{{k\left( { - 3} \right) + 1\left( 6 \right)}}{{k + 1}}$ and $6 = \dfrac{{k\left( {10} \right) + 1\left( { - 8} \right)}}{{k + 1}}$
We will simplify these equations to get the ratio in which the point $\left( { - 1,6} \right)$ divide the line segment joining the points $\left( { - 3,10} \right)$ and $\left( {6, - 8} \right)$.
Multiplying the terms in the numerator of the equation $- 1 = \dfrac{{k\left( { - 3} \right) + 1\left( 6 \right)}}{{k + 1}}$, we get
$\Rightarrow - 1 = \dfrac{{ - 3k + 6}}{{k + 1}}$
Multiplying both sides by $k + 1$, we get
$\Rightarrow - 1\left( {k + 1} \right) = - 3k + 6$
Multiplying the terms using the distributive law of multiplication, we get
$\Rightarrow - k - 1 = - 3k + 6$
Adding 1 on both sides, we get
$\begin{array}{l} \Rightarrow - k - 1 + 1 = - 3k + 6 + 1\\\ \Rightarrow - k = - 3k + 7\end{array}$
Adding $3k$ on both sides, we get
$\begin{array}{l} \Rightarrow - k + 3k = - 3k + 7 + 3k\\\ \Rightarrow 2k = 7\end{array}$
Dividing both sides by 2, we get
$\begin{array}{l} \Rightarrow k = \dfrac{7}{2}\\\ \Rightarrow \dfrac{k}{1} = \dfrac{7}{2}\end{array}$
Therefore, we get the ratio in which the point $\left( { - 1,6} \right)$ divide the line segment joining the points $\left( { - 3,10} \right)$ and $\left( {6, - 8} \right)$, as $7:2$.
The given statement is incorrect.
Thus, the correct option is option (b) False.

Note: We used the terms ‘abscissa’ and ‘ordinate’ in the solution. A point in the XY plane can be written as $\left( {x,y} \right)$. Here, the abscissa of the point is $x$, and the ordinate of the point is $y$.
We can verify the ratio $7:2$ by simplifying the equation formed by comparing the ordinates in the section formula.
Multiplying the terms in the numerator of the equation $6 = \dfrac{{k\left( {10} \right) + 1\left( { - 8} \right)}}{{k + 1}}$, we get
$\Rightarrow 6 = \dfrac{{10k - 8}}{{k + 1}}$
Multiplying both sides of the equation by $k + 1$, we get
$\Rightarrow 6\left( {k + 1} \right) = 10k - 8$
Multiplying the terms using the distributive law of multiplication, we get
$\Rightarrow 6k + 6 = 10k - 8$
Adding 8 on both sides, we get
$\begin{array}{l} \Rightarrow 6k + 6 + 8 = 10k - 8 + 8\\\ \Rightarrow 6k + 14 = 10k\end{array}$
Subtracting $6k$ from both sides of the equation, we get
$\begin{array}{l} \Rightarrow 6k + 14 - 6k = 10k - 6k\\\ \Rightarrow 14 = 4k\end{array}$
Dividing both sides by 4, we get
$\Rightarrow k = \dfrac{{14}}{4}$
Simplifying the expression, we get
$\begin{array}{l} \Rightarrow k = \dfrac{7}{2}\\\ \Rightarrow \dfrac{k}{1} = \dfrac{7}{2}\end{array}$
Hence, we have verified that the ratio in which the point $\left( { - 1,6} \right)$ divide the line segment joining the points $\left( { - 3,10} \right)$ and $\left( {6, - 8} \right)$, as $7:2$.