Find the coordinates of a point B where AB is the diameter of a circle
whose centre is (1, -3) and A(4,-1)
1
Answers
Answer:
The coordinates of the point B are ( - 2, - 5 ).
Step-by-step-explanation:
We have given the coordinates of the centre and one point of the diameter of the circle.
We have to find the coordinates of the other point of the diameter.
Let point O be the centre of the circle.
O ≡ ( 1, - 3 ) ≡ ( x, y )
A ≡ ( 4, - 1 ) ≡ ( x₁, y₁ )
B ≡ ( x₂, y₂ )
Now, AB is the diameter of the circle with centre O.
∴ O is the midpoint of AB.
∴ By midpoint formula,
x = ( x₁ + x₂ ) / 2 , y = ( y₁ + y₂ ) / 2
⇒ 1 = ( 4 + x₂ ) / 2 , - 3 = [ ( - 1 ) + y₂ ] / 2
⇒ 1 * 2 = 4 + x₂ , - 3 * 2 = - 1 + y₂
⇒ x₂ + 4 = 2 , y₂ - 1 = - 6
⇒ x₂ = 2 - 4 , y₂ = - 6 + 1
⇒ x₂ = - 2 , y₂ = - 5
∴ B ≡ ( - 2, - 5 )
∴ The coordinates of the point B are ( - 2, - 5 ).
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Additional Information:
1. Distance Formula:
The formula which is used to find the distance between two points using their coordinates is called distance formula.
- d ( A, B ) = √[ ( x₁ - x₂ )² + ( y₁ - y₂ )² ]
2. Section Formula:
The formula which is used to find the coordinates of a point which divides a line segment in a particular ratio is called section formula.
- x = ( mx₂ + nx₁ ) / ( m + n )
- y = ( my₂ + ny₁ ) / ( m + n )
3. Midpoint Formula:
The formula which is used to find the coordinates of the midpoint of a line segment is called midpoint formula.
- x = ( x₁ + x₂ ) / 2
- y = ( y₁ + y₂ ) / 2