Find the ratio in which the line segment joining the points and is divided by .

Section Formula: If a point divides the line segment joining and in the ratio internally, then: Here we already know P, A, and B — so we substitute the known coordinates and solve for the ratio . We will use the section formula to find the ratio. Step 1 — Identify the points Let the first point be . Its coordinates are . Let the second point be . Its coordinates are . Let the dividing point be . Its coordinates are . Let the ratio be . Step 2 — Use the x-coordinates We use the section formula for the x-coordinate. The formula is . Let's plug in our values. > Step 3 — Verify with y-coordinates We can also use the y-coordinate to check our answer. The formula is . Let's plug in our values. > Both calculations give the same value for . So, the ratio is . Answer > (i) The ratio is .

Coordinate Geometry | Exercise 7.2

Question 4

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

Solution

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:

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

Here we already know P, A, and B — so we substitute the known coordinates and solve for the ratio $m:n$ .

We will use the section formula to find the ratio.

Step 1 — Identify the points

Let the first point be $A$ . Its coordinates are $(x_1, y_1) = \mathbf{(-3, 10)}$ . Let the second point be $B$ . Its coordinates are $(x_2, y_2) = \mathbf{(6, -8)}$ . Let the dividing point be $P$ . Its coordinates are $(x, y) = \mathbf{(-1, 6)}$ . Let the ratio be $k:1$ .

Step 2 — Use the x-coordinates

We use the section formula for the x-coordinate. The formula is $x = \frac{k x_2 + 1 \cdot x_1}{k + 1}$ . Let's plug in our values.

$-1 = \frac{k(\mathbf{6}) + 1(\mathbf{-3})}{k + 1}$

$-1(k + 1) = 6k - 3$

$-k - 1 = 6k - 3$

$-1 + 3 = 6k + k$

$2 = 7k$

$\boxed{k = \frac{2}{7}}$

Step 3 — Verify with y-coordinates

We can also use the y-coordinate to check our answer. The formula is $y = \frac{k y_2 + 1 \cdot y_1}{k + 1}$ . Let's plug in our values.

$6 = \frac{k(\mathbf{-8}) + 1(\mathbf{10})}{k + 1}$

$6(k + 1) = -8k + 10$

$6k + 6 = -8k + 10$

$6k + 8k = 10 - 6$

$14k = 4$

$k = \frac{4}{14}$

$\boxed{k = \frac{2}{7}}$

Both calculations give the same value for $k$ . So, the ratio is $\mathbf{2:7}$ .

Answer

(i) The ratio is $\mathbf{2:7}$ .

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)]