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

Hint: 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 \[7:2\].