Question

the coordinates of one endpoint of a diameter of a circle are (4,-1 )and and the coordinates of the centre are(1,-3) then the coordinates of the and of the diameter is​

Answer 1

Answer:

The coordinates of the other endpoint of the diameter are ( - 2, - 5 ).

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

We have given the coordinates of one endpoint of diameter and centre of a circle.

We have to find the coordinates of the other endpoint of the diameter.

In figure,

Point O is the centre of the circle.

Seg AB is the diameter.

O ≡ ( 1, - 3 ) ≡ ( x, y )

A ≡ ( 4, - 1 ) ≡ ( x₁, y₁ )

B ≡ ( x₂, y₂ )

Now, we know that,

Centre of the circle is midpoint of the diameter.

∴ Point O is the midpoint of seg 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 other endpoint of the diameter 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 a point which divides a line segment in two equal parts is called midpoint formula.

• x = ( x₁ + x₂ ) / 2

• y = ( y₁ + y₂ ) / 2

Answer 2

Given ,

• The coordinates of one endpoint of a diameter of a circle are (4 , -1 )

• The centre of a circle is (1 , -3)

We know that ,

The centre of circle is the mid point of its diameter

And mid point formula is given by

$\boxed{ \tt{x = \frac{ x_{2} + x_{1}}{2} \: , \: y= \frac{ y_{2} + y_{1}}{2} }}$

Thus ,

$\tt \implies 1 = \frac{x + 4}{2} \: , \: - 3 = \frac{y + ( - 1)}{2}$

$\tt \implies 2 = x + 4 \: \: , \: \: y = - 6 = y - 1$

$\tt \implies x = - 2 \: , \: y = - 5$

Hence , the other end of diameter of circle is (-2 , -5)