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 yaxis which is equidistant from the points (5, − 2) and (− 3, 2).
Sol. ∵ Let P is on the yaxis
∴ 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 xaxis which is equidistant from the points (2, − 5) and (− 2, 9).
Sol. ∵ The required point ‘P’ is on xaxis.
∴ Coordinates of P are (x, 0).
∴ We have
AP = PB
⇒ AP^{2} = PB^{2}
⇒ (2 − x)2 + (− 5 + 0)2 =(− 2 − x)^{2} + (9 − 0)^{2}
⇒ 4 − 4x + x^{2} + 25 = 4 + 4x + x^{2} + 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 coordinates 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(x_{1}, y_{1}) and Q(x_{2}, y_{2})
∴ Using section formula, we have:
Thus, the coordinates of the required points are (0, −4) and (−2, 5)
Q23. If the midpoint 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 coordinates 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 coordinates of P be x and y.
∴ Using the section formula i.e.,
Next, Q divides AB in the ratio 2 : 3
∴ Coordinates of Q are :
Now, R divides AB in the ratio 3 : 2
⇒ Coordinates of R are :
The coordinates of P, Q and R are respectively:(2, 3), (3, 4) and (4, 5).
1. What is coordinate geometry? 
2. How is a point represented in coordinate geometry? 
3. What is the equation of a straight line in coordinate geometry? 
4. How do you find the distance between two points in coordinate geometry? 
5. How can we determine if three points are collinear in coordinate geometry? 

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