# Find the coordinates of the point P on the line segment joining $A(1,2)$ and $B(6,7)$ such that ${\text{AP = }}\dfrac{2}{5}{\text{AB}}{\text{.}}$

Answer

Verified

327.9k+ views

Hint-In order to solve such a question we will simply use a section formula which tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m:n.

$\left[ {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right]$

Where m, n is the ratios and x, y is the coordinates.

“Complete step-by-step answer:”

As we know that section formula or required coordinates of the point is given as

$\left[ {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right]$

Given that the coordinates of the line segment $A(1,2)$ and $B(6,7)$

And the relation as

${\text{AP = }}\dfrac{2}{5}{\text{AB}}{\text{.}}$

Let the coordinates of point be \[P\left( {x,y} \right)\] then

Here, point P is on AB such that ${\text{AP = }}\dfrac{2}{5}{\text{AB}}{\text{.}}$

$

\Rightarrow \dfrac{{{\text{AP}}}}{{{\text{AB}}}} = \dfrac{2}{5} \\

\Rightarrow 5{\text{AP = 2AB }}\left[ {\because {\text{ AB = AP + PB}}} \right] \\

\Rightarrow {\text{5AP = 2(AP + PB)}} \\

\Rightarrow {\text{5AP = 2AP + 2PB}} \\

\Rightarrow {\text{3AP = 2PB}} \\

\Rightarrow \dfrac{{{\text{AP}}}}{{{\text{PB}}}} = \dfrac{2}{3} \\

$

This means P divides AB in the ratio 2:3

As, we know that the section formula for required coordinate of the point is given as \[ \Rightarrow \left( {\dfrac{{{m_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\]

The ratio in which point P divides the line is 2:3,

Thus m=2, n=3

And the line points coordinates are $A(1,2)$ and $B(6,7)$

Therefore coordinates of P will be

\[

\Rightarrow \left( {\dfrac{{{m_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right) \\

\Rightarrow \left( {\dfrac{{2 \times 6 + 3 \times 1}}{{2 + 3}},\dfrac{{2 \times 7 + 3 \times 2}}{{2 + 3}}} \right) \\

\Rightarrow \left( {\dfrac{{15}}{5},\dfrac{{20}}{5}} \right) \\

\Rightarrow (3,4) \\

\]

Hence, the coordinates of the point which divides the line segment joining $(1,2)$ and B $(6,7)$ internally in the ratio $2:3$ is \[\left( {3,4} \right)\].

Note- To solve these types of problems remember all the formulas of coordinate geometry. And try to draw a rough sketch of the diagram on the paper, this helps a lot in solving the question. This problem can also be done by graphical method but coordinate geometry method is always the easiest and less time consuming method.

$\left[ {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right]$

Where m, n is the ratios and x, y is the coordinates.

“Complete step-by-step answer:”

As we know that section formula or required coordinates of the point is given as

$\left[ {\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right]$

Given that the coordinates of the line segment $A(1,2)$ and $B(6,7)$

And the relation as

${\text{AP = }}\dfrac{2}{5}{\text{AB}}{\text{.}}$

Let the coordinates of point be \[P\left( {x,y} \right)\] then

Here, point P is on AB such that ${\text{AP = }}\dfrac{2}{5}{\text{AB}}{\text{.}}$

$

\Rightarrow \dfrac{{{\text{AP}}}}{{{\text{AB}}}} = \dfrac{2}{5} \\

\Rightarrow 5{\text{AP = 2AB }}\left[ {\because {\text{ AB = AP + PB}}} \right] \\

\Rightarrow {\text{5AP = 2(AP + PB)}} \\

\Rightarrow {\text{5AP = 2AP + 2PB}} \\

\Rightarrow {\text{3AP = 2PB}} \\

\Rightarrow \dfrac{{{\text{AP}}}}{{{\text{PB}}}} = \dfrac{2}{3} \\

$

This means P divides AB in the ratio 2:3

As, we know that the section formula for required coordinate of the point is given as \[ \Rightarrow \left( {\dfrac{{{m_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right)\]

The ratio in which point P divides the line is 2:3,

Thus m=2, n=3

And the line points coordinates are $A(1,2)$ and $B(6,7)$

Therefore coordinates of P will be

\[

\Rightarrow \left( {\dfrac{{{m_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}}} \right) \\

\Rightarrow \left( {\dfrac{{2 \times 6 + 3 \times 1}}{{2 + 3}},\dfrac{{2 \times 7 + 3 \times 2}}{{2 + 3}}} \right) \\

\Rightarrow \left( {\dfrac{{15}}{5},\dfrac{{20}}{5}} \right) \\

\Rightarrow (3,4) \\

\]

Hence, the coordinates of the point which divides the line segment joining $(1,2)$ and B $(6,7)$ internally in the ratio $2:3$ is \[\left( {3,4} \right)\].

Note- To solve these types of problems remember all the formulas of coordinate geometry. And try to draw a rough sketch of the diagram on the paper, this helps a lot in solving the question. This problem can also be done by graphical method but coordinate geometry method is always the easiest and less time consuming method.

Last updated date: 02nd Jun 2023

•

Total views: 327.9k

•

Views today: 5.84k

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

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

Tropic of Cancer passes through how many states? Name them.

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

Name the Largest and the Smallest Cell in the Human Body ?