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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.
Solution :
Given:
The coordinates of point $A$ are $(x_1, y_1) = (-2, -2)$.
The coordinates of point $B$ are $(x_2, y_2) = (2, -4)$.
Point $P(x, y)$ lies on the line segment $AB$ such that $AP = \frac{3}{7} AB$.
To find:
The coordinates of point $P(x, y)$.
Step 1: Determining the ratio $AP : PB$
We are given $AP = \frac{3}{7} AB$.
Since $P$ lies on $AB$, we can write $AB = AP + PB$.
Substituting the given relation into the equation:
$AP = \frac{3}{7} (AP + PB)$
$7 AP = 3 AP + 3 PB$ [Multiplying both sides by 7]
$7 AP - 3 AP = 3 PB$ [Transposing $3 AP$ to the left side]
$4 AP = 3 PB$
$\frac{AP}{PB} = \frac{3}{4}$
Thus, point $P$ divides the line segment $AB$ in the ratio $m : n = 3 : 4$.
Step 2: Applying the Section Formula
The Section Formula states that the coordinates of a point $P(x, y)$ dividing the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m : n$ are given by:
$x = \frac{mx_2 + nx_1}{m + n}$
$y = \frac{my_2 + ny_1}{m + n}$
Step 3: Calculating the x-coordinate
$x = \frac{3(2) + 4(-2)}{3 + 4}$
$x = \frac{6 - 8}{7}$
$x = -\frac{2}{7}$
Step 4: Calculating the y-coordinate
$y = \frac{3(-4) + 4(-2)}{3 + 4}$
$y = \frac{-12 - 8}{7}$
$y = -\frac{20}{7}$
Conclusion:
The coordinates of point $P$ are $(-\frac{2}{7}, -\frac{20}{7})$.
Final Answer: The coordinates of point P are $\left(-\frac{2}{7}, -\frac{20}{7}\right)$.
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).
- 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.
- 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).
- 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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