Class 10 Maths Chapter 7 Question Answers - Coordinate Geometry - 2

Q12. Show that the triangle PQR formed by the points  and is an equilateral triangle. 

OR

Name the type of triangle PQR formed by the points   and  

Q13. The line joining the points (2, − 1) and (5, − 6) is bisected at P. If P lies on the line 2x + 4y + k = 0, find the value of k.
Sol. We have A (2, − 1) and B (5, − 6).
∵ P is the mid point of AB,
∴ Coordinates of P are:

Since P lies on the line 2x + 4y + k = 0
∴ We have:

Q14. Find the point on y-axis which is equidistant from the points (5, − 2) and (− 3, 2).
Sol. ∵ Let P is on the y-axis
∴ Coordinates of P are: (0, y)

Q15. The line joining the points (2, 1) and (5, − 8) is trisected at the points P and Q. If point P lies on the line 2x − y + k = 0, find the value of k.
Sol.

Q.16. Find the point on x-axis which is equidistant from the points (2, − 5) and (− 2, 9).
Sol. ∵ The required point ‘P’ is on x-axis.
∴ Coordinates of P are (x, 0).
∴ We have

AP = PB
⇒ AP² = PB²
⇒ (2 − x)2 + (− 5 + 0)2 =(− 2 − x)² + (9 − 0)²
⇒ 4 − 4x + x² + 25 = 4 + 4x + x² + 81
⇒ 4x + 25 = 4x + 81
⇒ − 8x = 56

∴ The required point is (−7, 0).

Q17. The line segment joining the points P (3, 3) and Q (6, − 6) is trisected at the points A and B such that A is nearer to P. It also lies on the line given by 2x + y + k = 0. Find the value of k.

Sol. ∵ PQ is trisected by A such that

Q18. Find the ratio in which the points (2, 4) divides the line segment joining the points A (− 2, 2) and B (3, 7). Also find the value of y.
Sol. Let P (2, y) divides the join of A (− 2, 2) and B (3, 7) in the ratio k:1
∴ Coordinates of P are:

Q19. Find the ratio in which the point (x, 2) divides the line segment joining the points (− 3, − 4) and (3, 5). Also find the value of x.

Sol. Let the required ratio = k : 1

∴ Coordinates of the point P are:

But the coordinates of P are (x, 2)

Q20. If P (9a – 2, –b) divides the line segment joining A (3a + 1, −3 ) and B (8a, 5) in the ratio 3 : 1, find the values of a and b.
Sol. ∵ P divides AB in the ratio 3 : 1
∴ Using the section formula, we have:

− b = 3 or b = −3
⇒ 36a − 8= 27a + 1 and b= −3
⇒ 9a = 9 and b = −3

Thus, the required value of a = 1 and b = −3

Q21. Find the ratio in which the point (x, − 1) divides the line segment joining the points (− 3, 5) and (2, − 5). Also find the value of x. 
Sol. Let the required ratio is k : 1

Q22. Find the co-ordinates of the points which divide the line segment joining A(2, −3) and B(−4, −6) into three equal parts.
Sol. Let the required points are P(x₁, y₁) and Q(x₂, y₂)
 ∴ Using section formula, we have:

Thus, the coordinates of the required points are (0, −4) and (−2, 5)

Q23. If the mid-point of the line segment joining the point A(3, 4) and B(k, 6) is P(x, y) and x + y – 10 = 0, then find the value of k.
Sol. ∵ Mid point of the line segment joining A(3, 4) and B(k, 6)

Q24. Point P, Q, R and S divide the line segment joining the points A (1, 2) and B (6, 7) in 5 equal parts. Find the co-ordinates of the points P, Q and R.

∴  P, Q, R and S, divide AB into five      equal parts.
∴ AP = PQ = QR = RS = SB
Now, P divides AB in the ratio 1 : 4    
Let, the co-ordinates of P be x and y.
∴ Using the section formula i.e.,


Next, Q divides AB in the ratio 2 : 3
∴ Co-ordinates of Q are :

Now, R divides AB in the ratio 3 : 2
⇒  Co-ordinates of R are :

The co-ordinates of P, Q  and R are respectively:(2, 3), (3, 4) and (4, 5).