Find the centre of a circle passing through the points (6, - 6), (3, - 7) and (3, 3)
Solution:
The distance between the two points can be measured using the Distance Formula which is given by:
Distance Formula = √( x₂ - x₁ )2 + (y₂ - y₁)2
Let's draw the required figure.
According to the diagram,
Let O(x,y) be the centre of the circle.
Let the points (6,- 6), (3, -7), and (3, 3) represent the points A, B, and C on the circumference of the circle.
Distance from centre O to A, B, C are found below using the Distance formula.
Hence, OA = √ [(x - 6)2 + (y + 6)2]
OB = √ [(x - 3)2 + (y + 7)2]
OC = √ [(x - 3)2 + (y - 3)2]
From the figure ,
OA = OB (radii of the same circle)
√ [(x - 6)2 + (y + 6)2] = √ [(x - 3)2 + (y + 7)2]
x2 + 36 - 12x + y2 + 36 + 12y = x2 + 9 - 6x + y2 + 49 + 14y (Squaring on both sides)
- 6x - 2y + 14 = 0
3x + y = 7 ..... (1)
Similarly, OA = OC (radii of the same circle)
√ [(x - 6)2 + (y + 6)2] = √ [(x - 3)2 + (y - 3)2]
x2 + 36 - 12x + y2 + 36 + 12y = x2 + 9 - 6x + y2 + 9 - 6y (Squaring on both sides)
- 6x + 18y + 54 = 0
- 3x + 9y = - 27 ..... (2)
On adding Equation (1) and (2), we obtain
3x + y - 3x + 9y = 7 + (-27)
10y = - 20
y = - 2
From Equation (1), we obtain
3x - 2 = 7
3x = 9
x = 3
Therefore, the centre of the circle is (3, - 2).
☛ Check: NCERT Solutions for Class 10 Maths Chapter 7
Video Solution:
Find the centre of a circle passing through the points (6, - 6), (3, - 7) and (3, 3).
Maths NCERT Solutions Class 10 Chapter 7 Exercise 7.4 Question 3
Summary:
The centre of a circle passing through the points (6, - 6), (3, - 7) and (3, 3) is (3, - 2).
☛ Related Questions:
- The two opposite vertices of a square are (- 1, 2) and (3, 2). Find the coordinates of the other two vertices.
- The vertices of a ∆ABC are A (4, 6), B (1, 5) and C (7, 2). A line is drawn to intersect sides AB and AC at D and E respectively, such that AD/AB = AE/AC = 1/4.Calculate the area of the ∆ADE and compare it with the area of ∆ABC. (Recall Theorem 6.2 and Theorem 6.6).
- Let A (4, 2), B(6, 5) and C(1, 4) be the vertices of ∆ABC.(i) The median from A meets BC at D. Find the coordinates of the point D.(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.(iv) What do yo observe? [Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio 2 : 1.]