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

means P is ths of the way from A to B, so AP takes 3 parts and PB takes the remaining 4 parts — giving a ratio of . Section Formula: If a point divides the line segment joining and in the ratio internally, then: We will use the section formula to find point P. Step 1 — Find the ratio The problem tells us . This means AP is 3 parts of the segment. AB is 7 parts in total. So, PB must be parts. Point P divides AB in the ratio . We can say and . > Step 2 — Identify coordinates Let's write down the given points. Point A is . Point B is . Step 3 — Calculate x-coordinate We use the section formula for . The formula is . > Step 4 — Calculate y-coordinate Now we find the y-coordinate. The formula is . > Step 5 — State P's coordinates Let's write the coordinates of P. P has coordinates . > Answer > The coordinates of P are .

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 $\text{AP} = \frac{3}{7} \text{AB}$ and P lies on the line segment AB.

Solution

$\text{AP} = \frac{3}{7}\text{AB}$ means P is $\frac{3}{7}$ ths of the way from A to B, so AP takes 3 parts and PB takes the remaining 4 parts — giving a ratio of $3:4$ .

Section Formula: If a point $\text{P}(x, y)$ divides the line segment joining $\text{A}(x_1, y_1)$ and $\text{B}(x_2, y_2)$ in the ratio $m:n$ internally, then:

$x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}$

We will use the section formula to find point P.

Step 1 — Find the ratio

The problem tells us $\text{AP} = \frac{3}{7} \text{AB}$ . This means AP is 3 parts of the segment. AB is 7 parts in total. So, PB must be $7 - 3 = 4$ parts. Point P divides AB in the ratio $3:4$ . We can say $m = 3$ and $n = 4$ .

$\boxed{m:n = 3:4}$

Step 2 — Identify coordinates

Let's write down the given points. Point A is $(x_1, y_1) = (-2, -2)$ . Point B is $(x_2, y_2) = (2, -4)$ .

Step 3 — Calculate x-coordinate

We use the section formula for $x$ . The formula is $x = \frac{mx_2 + nx_1}{m+n}$ . $x = \frac{3(2) + 4(-2)}{3+4}$ $= \frac{6 - 8}{7}$ $= \frac{-2}{7}$

$\boxed{x = -\frac{2}{7}}$

Step 4 — Calculate y-coordinate

Now we find the y-coordinate. The formula is $y = \frac{my_2 + ny_1}{m+n}$ . $y = \frac{3(-4) + 4(-2)}{3+4}$ $= \frac{-12 - 8}{7}$ $= \frac{-20}{7}$

$\boxed{y = -\frac{20}{7}}$

Step 5 — State P's coordinates

Let's write the coordinates of P. P has coordinates $(x, y)$ .

$\boxed{\text{P} \left(-\frac{2}{7}, -\frac{20}{7}\right)}$

Answer

The coordinates of P are $\left(-\frac{2}{7}, -\frac{20}{7}\right)$ .

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 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. 7.12. Niharika runs $\frac{1}{4}$ th the distance AD on the 2nd line and posts a green flag. Preet runs $\frac{1}{5}$ 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 A, where AB is the diameter of a circle whose centre is $(2, -3)$ and B is $(1, 4)$ .

If A and B are $(-2, -2)$ and $(2, -4)$ , respectively, find the coordinates of P such that $\text{AP} = \frac{3}{7} \text{AB}$ 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 = $\frac{1}{2}$ (product of its diagonals)]