If A and B are and , respectively, find the coordinates of P such that and P lies on the line segment AB.

Understand the Question Given points and , with lying on the line segment such that . To find the coordinates of , we first determine the ratio in which divides the segment . We then apply the section formula: Step 1 · Find the Ratio Given .Since lies on : Therefore, the ratio is: So, divides internally in the ratio . Step 2 · Calculate Coordinates of P Using Section Formula Here, , , , and . Using the section formula: Therefore, the coordinates of are . Answer > Common Mistakes Ratio Error: Using the ratio directly instead of calculating . Sign Errors in Coordinates: Forgetting the negative signs while substituting or into the section formula.

Coordinate Geometry | Exercise 7.2

Question 8

If A and B are $(-2, -2)$ and $(2, -4)$ , respectively, find the coordinates of P such that $\text{AP} = \dfrac{3}{7} \text{AB}$ and P lies on the line segment AB.

Solution

Understand the Question

Given points $\text{A}(-2, -2)$ and $\text{B}(2, -4)$ , with $\text{P}$ lying on the line segment $\text{AB}$ such that $\text{AP} = \dfrac{3}{7} \text{AB}$ .
To find the coordinates of $\text{P}$ , we first determine the ratio $\text{AP} : \text{PB}$ in which $\text{P}$ divides the segment $\text{AB}$ .
We then apply the section formula: $x = \dfrac{mx_2 + nx_1}{m + n}, \quad y = \dfrac{my_2 + ny_1}{m + n}$

Step 1 · Find the Ratio $\text{AP} : \text{PB}$

Given $\text{AP} = \dfrac{3}{7} \text{AB}$ .Since $\text{P}$ lies on $\text{AB}$ :

\begin{aligned} \text{PB} &= \text{AB} - \text{AP} \\ &= \text{AB} - \dfrac{3}{7} \text{AB} \\ &= \dfrac{4}{7} \text{AB} \end{aligned}

Therefore, the ratio is: $\dfrac{\text{AP}}{\text{PB}} = \dfrac{\frac{3}{7}\text{AB}}{\frac{4}{7}\text{AB}} = \dfrac{3}{4}$

So, $\text{P}$ divides $\text{AB}$ internally in the ratio $m : n = 3 : 4$ .

Step 2 · Calculate Coordinates of P Using Section Formula

Here, $(x_1, y_1) = (-2, -2)$ , $(x_2, y_2) = (2, -4)$ , $m = 3$ , and $n = 4$ .

Using the section formula:

\begin{aligned} x &= \dfrac{mx_2 + nx_1}{m + n} \\[0.6em] &= \dfrac{3(2) + 4(-2)}{3 + 4} \\[0.6em] &= \dfrac{6 - 8}{7} \\[0.6em] &= -\dfrac{2}{7} \end{aligned}

\begin{aligned} y &= \dfrac{my_2 + ny_1}{m + n} \\[0.6em] &= \dfrac{3(-4) + 4(-2)}{3 + 4} \\[0.6em] &= \dfrac{-12 - 8}{7} \\[0.6em] &= -\dfrac{20}{7} \end{aligned}

Therefore, the coordinates of $\text{P}$ are $\left(-\dfrac{2}{7}, -\dfrac{20}{7}\right)$ .

Answer

$\left(-\dfrac{2}{7}, -\dfrac{20}{7}\right)$

Common Mistakes

Ratio Error: Using the ratio $3:7$ directly instead of calculating $\text{AP} : \text{PB} = 3 : (7 - 3) = 3 : 4$ .
Sign Errors in Coordinates: Forgetting the negative signs while substituting $(x_1, y_1) = (-2, -2)$ or $(x_2, y_2) = (2, -4)$ into the section formula.

More questions in Exercise 7.2

Find the coordinates of the point which divides the join of $(-1, 7)$ and $(4, -3)$ in the ratio $2 : 3$ .

Find the coordinates of the points of trisection of the line segment joining $(4, -1)$ and $(-2, -3)$ .

To conduct Sports Day activities, in your rectangular shaped school ground $ABCD$ , lines have been drawn with chalk powder at a distance of $1\text{ m}$ each. $100$ flower pots have been placed at a distance of $1\text{ m}$ from each other along $AD$ , as shown in Fig. 7.12. Niharika runs $\dfrac{1}{4}\text{th}$ the distance $AD$ on the 2nd line and posts a green flag. Preet runs $\dfrac{1}{5}\text{th}$ the distance $AD$ on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Find the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by $(-1, 6)$ .

Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$ -axis. Also find the coordinates of the point of division.

If $(1, 2)$ , $(4, y)$ , $(x, 6)$ and $(3, 5)$ are the vertices of a parallelogram taken in order, find $x$ and $y$ .

Find the coordinates of a point $\text{A}$ , where $\text{AB}$ is the diameter of a circle whose centre is $(2, -3)$ and $\text{B}$ is $(1, 4)$ .

If A and B are $(-2, -2)$ and $(2, -4)$ , respectively, find the coordinates of P such that $\text{AP} = \dfrac{3}{7} \text{AB}$ and P lies on the line segment AB.

Find the coordinates of the points which divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four equal parts.

Q10

Find the area of a rhombus if its vertices are $(3, 0)$ , $(4, 5)$ , $(-1, 4)$ and $(-2, -1)$ taken in order.

[Hint : Area of a rhombus = $\dfrac{1}{2}$ (product of its diagonals)]