Question

If the first point of trisection of AB is \[\left( {t,2t} \right)\] and the ends A, B moves on x and y axis respectively, then locus of midpoint of AB is
A.\[x = y\]
B.\[2x = y\]
C.\[4x = y\]
D.\[x = 4y\]

Answer 1

Hint: Here, we will divide a line segment into three parts. Then by using the section formula and the given co-ordinates we will find the coordinates of the line segment. Then by equating the co-ordinates of P, we will find the locus of midpoint of AB.

Formula Used:
Section formula: \[\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\] where \[m:n\] is the ratio of a point dividing the line segment and \[\left( {{x_1},{y_1}} \right)\] , \[\left( {{x_2},{y_2}} \right)\] are the coordinates of the line segment.

Complete step-by-step answer:
 We will first draw the diagram based on the given information.

Let P and M be the points of Trisection of the line segment joining the points A\[\left( {2h,0} \right)\] and B\[\left( {0,2k} \right)\] such that P is nearer to A. Let P\[\left( {t,2t} \right)\] and M\[\left( {h,k} \right)\] be the co-ordinates of the points P and M.
Therefore, P divides the line segment in the ratio \[1:2\] and Q divides the line segment in the ratio \[2:1\].
By using the section formula \[\left( {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\], we get
\[ \Rightarrow \] Co-ordinates of P\[ = \left( {\dfrac{{1\left( 0 \right) + 2\left( {2h} \right)}}{{1 + 2}},\dfrac{{1\left( {2k} \right) + 2\left( 0 \right)}}{{1 + 2}}} \right)\]
Multiplying the terms, we get
\[ \Rightarrow \left( {t,2t} \right) = \left( {\dfrac{{4h}}{3},\dfrac{{2k}}{3}} \right)\]
\[ \Rightarrow \left( {\dfrac{{4h}}{3},\dfrac{{2k}}{3}} \right) = \left( {t,2t} \right)\]
By comparing the co-ordinates of \[x\] and \[y\], we get
\[ \Rightarrow \dfrac{{4h}}{3} = t\] and \[\dfrac{{2k}}{3} = 2t\]
\[ \Rightarrow 4h = 3t\] and \[2k = 6t\]
\[ \Rightarrow 4h = 3t\] and \[k = 3t\]
By substituting \[k = 3t\] in \[4h = 3t\], we get
\[4h = k\]
So, by substituting the co-ordinates, we get
\[ \Rightarrow 4x = y\]
Therefore, the locus of midpoint of AB is \[4x = y\] and thus Option (C) is the correct answer.

Note: We know that trisection is the division of a line segment into three equal parts. Section formula is used to determine the co-ordinate of a point that divides a line segment joining two points into two parts such that their ratio of length is \[m:n\]. The sectional formula can also be used to find the co-ordinate of a point that lies outside a circle.