Find the coordinates of the point which divides the join of and in the ratio .

Understand the Question Let be the point that divides the line segment joining and internally in the ratio . By the Section Formula, the coordinates of are given by: Step 1 · Calculate the x-coordinate Let the given points be and , with ratio .Using the section formula for Step 2 · Calculate the y-coordinate Using the section formula for Answer > Common Mistakes Ratio-Coordinate Mismatch: Multiplying by instead of . Remember that multiplies the coordinates of the second point and multiplies the coordinates of the first point . Sign Errors: Mishandling negative signs during substitution, e.g., writing instead of , or instead of .

Coordinate Geometry | Exercise 7.2

Question 1

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

Solution

Understand the Question

Let $P(x, y)$ be the point that divides the line segment joining $A(x_1, y_1) = (-1, 7)$ and $B(x_2, y_2) = (4, -3)$ internally in the ratio $m_1 : m_2 = 2 : 3$ .
By the Section Formula, the coordinates of $P(x, y)$ are given by: $x = \dfrac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \quad y = \dfrac{m_1 y_2 + m_2 y_1}{m_1 + m_2}$

Step 1 · Calculate the x-coordinate

Let the given points be $A(x_1, y_1) = (-1, 7)$ and $B(x_2, y_2) = (4, -3)$ , with ratio $m_1 : m_2 = 2 : 3$ .Using the section formula for $x$

\begin{aligned} x &= \dfrac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \\[0.6em] &= \dfrac{2 \times 4 + 3 \times (-1)}{2 + 3} \\[0.6em] &= \dfrac{8 - 3}{5} \\[0.6em] &= \dfrac{5}{5} = 1 \end{aligned}

Step 2 · Calculate the y-coordinate

Using the section formula for $y$

\begin{aligned} y &= \dfrac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \\[0.6em] &= \dfrac{2 \times (-3) + 3 \times 7}{2 + 3} \\[0.6em] &= \dfrac{-6 + 21}{5} \\[0.6em] &= \dfrac{15}{5} = 3 \end{aligned}

Answer

$(1, 3)$

Common Mistakes

Ratio-Coordinate Mismatch: Multiplying $m_1$ by $x_1$ instead of $x_2$ . Remember that $m_1$ multiplies the coordinates of the second point $(x_2, y_2)$ and $m_2$ multiplies the coordinates of the first point $(x_1, y_1)$ .
Sign Errors: Mishandling negative signs during substitution, e.g., writing $3 \times (-1) = 3$ instead of $-3$ , or $2 \times (-3) = 6$ instead of $-6$ .

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)]