Answer

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**Hint:**In this question, you have to find the point which is three-fourth of the way from A(3,1) to B(2,5). You need to consider a variable for the three-fourth point in the way from A(3,1) to B(2,5). You can use the section formula to solve this problem. So let us see how we can solve this problem.

**Complete Step by Step Solution:**

Given that there is a three-fourth point in between A(3,1) to B(2,5). Let S be the point which is three-fourth of the way from A(3,1) to B(2,5).

So we have, AP : AB = 3 : 4

Now, AB = AP + PB

$\therefore \dfrac{{AP}}{{AB}} = \dfrac{{AP}}{{(AP + PB)}} = \dfrac{3}{4}$

On cross multiplying we get,

$\Rightarrow 4AP = 3AP + 3BP$

Subtracting 3AP from both the sides of the above expression,

$\Rightarrow 4AP - 3AP = 3BP$

$\Rightarrow AP = 3BP$

$\Rightarrow \dfrac{{AP}}{{AB}} = \dfrac{3}{1}$

So, the ration m : n = 3 : 1

${x_1} = 3,{y_1} = 1,{x_2} = 2,{y_2} = 5$

By section formula we get: $x = \dfrac{{m{x_2} + n{x_1}}}{{m + n}}$

Putting all the values of the variable in the above expression we get,

$\Rightarrow x = \dfrac{{3 \times 2 + 1 \times 3}}{{3 + 1}}$

After addition and subtraction, we get,

$\Rightarrow x = \dfrac{9}{4}$

By section formula we get: $y = \dfrac{{m{y_2} + n{y_1}}}{{m + n}}$

$\Rightarrow y = \dfrac{{3 \times 5 + 1 \times 1}}{{3 + 1}}$

After addition and subtraction, we get,

$\Rightarrow y = \dfrac{2}{4}$

Therefore, $y = \dfrac{1}{2}$

**So the coordinate of P is $(\dfrac{9}{4},\dfrac{1}{2})$.**

**Note:**

In the above solution we have used the section formula. The sectional formula is $M(x,y) = (\dfrac{{m{x_2} + n{x_1}}}{{m + n}},\dfrac{{m{y_2} + n{y_1}}}{{m + n}})$. In the question, it is given that the point is three-fourth from A(3,1) to B(2,5). Section formula is used when we have to find the ratio in which the line segment is divided by a point externally or internally.

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