Answer

Verified

426.9k+ 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

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

Who was the Governor general of India at the time of class 11 social science CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference Between Plant Cell and Animal Cell