Coordinate Geometry Chapter 2 Internal And External Division Of Straight Line Segment
Concept of determination of a formula of coordinates of a point when a straight line segment is divided internally or externally in a given ratio.
1. Co-ordinates of the point P, which divides the line segment joining the points A (x1, y1) and B (x2, y2) internally in the ratio m:n are \(\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)\)
2. Co-ordinates of the point P, which divides the line segment joining the points A (x1, y1) and B (x2, y2) externally in the ratio m: n are \(\left(\frac{m x_2-n x_1}{m-n}, \frac{m y_2-n y_1}{m-n}\right)\)
Co-ordinate the midpoint P of the line segment joining two points A (x1, y1), B (x2, y2) are \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\)
Co-ordinates of the centroid of the triangle ABC where co-ordinates of A, B, C are (x1, y1),(x2, y2) and (x3, y3) respectively \(\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\)
Read and Learn More WBBSE Solutions For Class 9 Maths
Coordinate Geometry Chapter 2 Internal And External Division Of Straight Line Segment Fill In The Blanks
Example 1. The coordinates of the point which divides the line segment joining (6, – 4) and (-8, 10) in the ratio 3: 4 internally is _______
Solution: \(\left(0, \frac{-26}{7}\right)\)
Example 2. The coordinates of the midpoint of the line segment joining two points (5, 4), (3, -4) is _______
Solution: (4, 0).
Example 3. The ratio in which the point (1, 3) divides the line segment joining the points (4, 6), (3, 5) is ________ externally.
Solution: 32.
Example 4. The ratio at which the line segment joining the points (7, 3), (-9, 6) divides by Y axis is _______
Solution: 79.
Example 5. A (2, 3), B (9, 6), C (10, 12), and D (10, 12) are joined in order. ABCD is a _______
Solution: parallelogram.
Example 6. The points (3, 2), (6, 3), (x, y) and (6, 5) are joined in order to make a parallelogram (x, y) = _______
Solution: (9, 6).
Coordinate Geometry Chapter 2 Internal And External Division Of Straight Line Segment True Or False
Example 1. The Centroid of the triangle formed by the points (a – b, b – c), (-a, -b), (b, c) are (0, 0).
Solution: The statement is true.
Example 2. The coordinates of the point on the x axis at which the line segment joining the points (3, 4), (-3, -4) is bisected are (3, 0).
Solution: The statement is false.
Example 3. P is such a point on the line segment AB such that AP = PB. If A (-2, 4) and P (1, 5), then (4, 6).
Solution: The statement is true.
Example 4. The distance between two points on the x axis is 10 units. It origin the midpoint of these two points then the coordinates of the two point are (5, 0) and (-5, 0).
Solution: The statement is true.
Example 5. The coordinates of the point which divides the line segment joining the points (-1, 2), (2, -1) externally is 2 5 are (3, 4).
Solution: The statement is false.
Coordinate Geometry Chapter 2 Internal And External Division Of Straight Line Segment Short Answer Type Questions
Example 1. C is the centre of a circle and AB is the diameter, the coordinates of A and C are (6, -7) and (5, -2). Calculate the coordinates of B.
Solution: Let coordinates of B is (x, y)
∴ \(\frac{6+x}{2}=5 \Rightarrow x=4\)
\(\frac{-7+y}{2}=-2 \Rightarrow y=3,\)
∴ Co-ordinates are (4, 3).
∴ The coordinates of B is (4, 3).
Example 2. The points P and Q lie on 1st and 3rd quadrants respectively. The distance of the two points for x axis and y axis are 6 units, 4 units respectively. Find the midpoint of PQ.
Solution: Midpoint = \(\left(\frac{4-4}{2}, \frac{6-6}{2}\right)=(0,0)\)
The midpoint of PQ = \(\left(\frac{4-4}{2}, \frac{6-6}{2}\right)=(0,0)\)
Example 3. The point P lies on AB, AP = PB, the coordinates of A and B are (3, -4) and (-5, 2) respectively. Find their coordinates of P.
Solution: Coordinates of P = \(=\left(\frac{3-5}{2}, \frac{-h+2}{2}\right)=(-1,-1) .\)
Example 4. Points A and B lie on the 2nd and 4th quadrants. The distance of each point from x-axis and y-axis are 8 units and 6 units respectively. Write the coordinates of the midpoint of AB.
Solution: Midpoint of AB = \(\left(\frac{6+6}{2}, \frac{8-8}{2}\right)\) = (0, 0)
Example 5. The sides of a rectangle ABCD are parallel to the coordinates axes. Coordinates of B and D are (7, 3) and (2, 6). Write the coordinates of A, C and the midpoint of AC.
Solution: Absciss of C = Absciss of B = 7
∴ Co-ordinate of C is (7, 6)
⇒ Absciss of A = Absciss of A = 2
⇒ Ordinate of A = Ordinate of B = 3
⇒ Co-ordinates of A (2, 3)
⇒ Co-ordinates of midpoint of AC = \(\left(\frac{7+2}{2}, \frac{6+3}{2}\right)=\left(\frac{9}{2}, \frac{9}{2}\right)\)
Example 6. The coordinates of the vertices of a triangle are (4, -3), (-5, 2), (x, y). If the centroid of the triangle is at the origin, then find X, Y.
Solution: \(\frac{4-5+x}{3}=0 \Rightarrow x=1 \text { and } \frac{-3+2+y}{3}=0 \Rightarrow y=1\)
Example 7. Find the ratio in which the line segment joining the points (2, 3) and (5,- h) divided by x axis internally.
Solution: Let the coordinates at x-axis at which the line segment intersects i.e. (k, 0) and the ratio be m: n.
∴ \(\frac{-4 m+3 n}{m+n}\) =0 m: n = 3: 4
Example 8. Find the ratio in which the point (9, -23) divides the line segment joining (6, 4) and (7,-5)
Solution: Let the required ratio be m: n; 9 = \(\frac{7 n+6 n}{m+n}\) 9m + 9n = 7m + 6n
⇒ 2m = -3n, \(\frac{m}{n}=-\frac{3}{2}\)
∴ (9, -23) divides the line segment in 3: 2 externally.