Question

Find the value of x if the points A (4,3) and B(x,5) lie on a circle whose centre is O(2,3).

Answer 1

Given, A (4, 3) and B (x, 5) are two points on the circle. Centre of the circle is O (2, 3).

∴ OB = OA  (Radius of the circle)

⇒ OB2 = OA2

⇒ (x – 2)2 + (5 – 3)2 = (4 – 2)2 + (3 – 3)2 

⇒ (x – 2)2 + 4 = 4

⇒ (x – 2)2 = 0

⇒ x – 2 = 0

⇒ x = 2

Thus, the value of x is 2.

Answer 2

Coordinate geometry is the branch of mathematics which deals with the position of an object lying in a plane.

Each  point in cartesian plane has two coordinates X coordinate and Y coordinate.
The X co-ordinate is called the abscissa.
The Y co-ordinate is called the ordinate.

Coordinates X and Y taken together are called coordinates of a point. (x,y) is called an ordered pair.

DISTANCE FORMULA:

The distance between any two points A(x1,y1) and B(x2,y2) is:
AB=√(x1 - x2)² + (y1 - y2)²]

SOLUTION:

GIVEN:
A and B lie on the circle having centre O.

Here, A( 4, 3) , B( x , 5)  , O (2 , 3)

For distance OA :  x1 = 4 , y1 = 3 , x2 =2 & ​y2 = 3
For distance OB :  x1 = 2 , y1 = 3, x2 = x & ​y2 = 5

OA = OB     [Radius of a Circle]
√(4-2)² +(3-3)²  = √(2 -x )² +(3-5)²

[Distance Formula=√(x1 - x2)² + (y1 - y2)²]

On squaring both sides
(2)²  = (2 -x )² +(-2)²
4 = (2²+x²-2×2x) + 4

[(a-b)² = a² + b² -2ab]
4 = 4 +x² -4x +4
4 = x² - 4x +8
4 -8 = x² - 4x
-4 = x² - 4x
x² -4x +4= 0
x² -2x -2x +4= 0     [By factorization]
x(x-2) -2(x-2) = 0
(x-2) (x-2) = 0
(x-2) = 0   or (x-2) = 0
x = 2 or x= 2

Hence, the value of x is 2.

HOPE THIS WILL HELP YOU..