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Q5:
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.
Solution :
Given:
The coordinates of the endpoints of the line segment are $A(x_1, y_1) = (1, -5)$ and $B(x_2, y_2) = (-4, 5)$.
The line segment $AB$ is divided by the $x$-axis.
To Find:
1. The ratio ($k:1$) in which the $x$-axis divides the line segment $AB$.
2. The coordinates of the point of division ($P$).
Step 1: Defining the Point of Division
Let the point $P$ on the $x$-axis divide the line segment $AB$ in the ratio $k:1$.
Since the point $P$ lies on the $x$-axis, its $y$-coordinate must be $0$. Therefore, the coordinates of $P$ are $(x, 0)$.
Step 2: Applying the Section Formula
The Section Formula states that if a point $P(x, y)$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$, then:
$x = \frac{mx_2 + nx_1}{m+n}$ and $y = \frac{my_2 + ny_1}{m+n}$
Substituting our values ($m=k, n=1, x_1=1, y_1=-5, x_2=-4, y_2=5$):
$y = \frac{k(5) + 1(-5)}{k+1}$
Step 3: Solving for the Ratio $k$
Since the point lies on the $x$-axis, $y = 0$.
$0 = \frac{5k - 5}{k+1}$
Multiplying both sides by $(k+1)$ (assuming $k \neq -1$):
$0 = 5k - 5$
$5k = 5$
$k = \frac{5}{5} = 1$
Thus, the ratio $k:1$ is $1:1$.
Step 4: Finding the Coordinates of the Point of Division
Now, substitute $k=1$ into the formula for the $x$-coordinate:
$x = \frac{k(x_2) + 1(x_1)}{k+1}$
$x = \frac{1(-4) + 1(1)}{1+1}$
$x = \frac{-4 + 1}{2}$
$x = \frac{-3}{2} = -1.5$
The coordinates of the point of division $P$ are $(x, y) = \left(-\frac{3}{2}, 0\right)$.
Final Answer: The line segment is divided by the x-axis in the ratio 1:1, and the coordinates of the point of division are (-1.5, 0).
More Questions from Class 10 Mathematics Coordinate geometry EXERCISE 7.2
- Q1: Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3.
- 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)]
- Q2: Find the coordinates of the points of trisection of the line segment joining (4, –1) and (–2, –3).
- Q3: 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?
- Q4: Find the ratio in which the line segment joining the points (– 3, 10) and (6, – 8) is divided by (– 1, 6).
- Q6: If (1, 2), (4, $y$), ($x$, 6) and (3, 5) are the vertices of a parallelogram taken in order, find $x$ and $y$.
- Q7: 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).
- Q8: If A and B are (– 2, – 2) and (2, – 4), respectively, find the coordinates of P such that AP = $\frac{3}{7}$ AB and P lies on the line segment AB.
- Q9: Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.
CBSE Solutions for Class 10 Mathematics Coordinate geometry
Chapters in CBSE - Class 10 Mathematics
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