# If the first point of trisection of AB is (t, 2t) and the ends A, B move on ‘x’ and ‘y’ axis respectfully, then the focus of midpoint of AB is

A.x = y

B.2x = y

C.4x = y

D.x = 4y

Answer

Verified

175.8k+ views

**Hint**: We know the section formula, the coordinates of the point P(x, y) which divides the line segment joining the points \[A({x_1},{y_1})\] and \[B({x_2},{y_2})\] internally, in the ratio \[{m_1}:{m_2}\] are \[\left( {\dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}},\dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}}} \right)\] . We choose the midpoint as \[M(h,k)\] and using the above formula we find the coordinate values ‘h’ and ‘k’ to obtain the required result.

**:**

__Complete step-by-step answer__Let \[M(h,k)\] be the midpoint of line AB. Given A, B move on the ‘x’ and ‘y’ axis respectively. Then it is obvious that the coordinates of A and B are \[A(2h,0)\] and \[B(0,2k)\] .

Since point \[P(t,2t)\] is first trisection it divides A and B into \[1:2\] ratio. See in the below diagram for understanding point of view.

We have \[A({x_1},{y_1}) = A(2h,0)\] , \[B({x_2},{y_2}) = B(0,2k)\] and which divides in the ratio \[1:2\] . Using the section formula we find the coordinates of the point P.

Using the formula \[\left( {\dfrac{{{m_1}{x_2} + {m_2}{x_1}}}{{{m_1} + {m_2}}},\dfrac{{{m_1}{y_2} + {m_2}{y_1}}}{{{m_1} + {m_2}}}} \right)\]

\[ \Rightarrow = P\left( {\dfrac{{\left( {1 \times 0} \right) + \left( {2 \times 2h} \right)}}{{1 + 2}},\dfrac{{\left( {1 \times 2k} \right) + \left( {2 \times 0} \right)}}{{1 + 2}}} \right)\]

\[ \Rightarrow = P\left( {\dfrac{{0 + 4h}}{3},\dfrac{{2k + 0}}{3}} \right)\]

\[ \Rightarrow = P\left( {\dfrac{{4h}}{3},\dfrac{{2k}}{3}} \right)\]

But we already have point p coordinates as \[P(t,2t)\] .

Comparing ‘x’ and ‘y’ coordinates in both Points P we have

The ‘x’ coordinate \[\dfrac{{4h}}{3} = t\]

Multiply by 3 on both sides.

\[ \Rightarrow 4h = 3t{\text{ - - - - - - - (1)}}\]

The ‘y’ coordinate \[\dfrac{{2k}}{3} = 2t\]

Cancelling 2 on both sides,

\[ \Rightarrow \dfrac{k}{3} = t\]

Multiply by 3 on both sides,

\[ \Rightarrow k = 3t{\text{ - - - - - - - (2)}}\]

Now substituting equation (2) in equation (1) we have \[ \Rightarrow 4h = k\] .

But we have options in ‘x’ and ‘y’ variables, so we have

\[ \Rightarrow 4x = y\] (Because \[(h,y) = (x,y)\] )

**So, the correct answer is “Option C”.**

**Note**: In general Trisection is the division of a quantity or figure into three equal parts. In the above problem when P is the first trisection there is another two trisection. Hence we take the ratio as \[1:2\] . Remember the formula of section formula. Careful in the substitution and calculation part.

Recently Updated Pages

Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE