Question

What are the coordinates of the centre of a circle. where coordinates of end points of diameter are given as (9,9) and (9,-21)?​

Answer 1

Given :

• Coordinates of end points of diameter are (9,9) and (9,-21)

To Find :

• The coordinates of centre of the circle

Figure :

$\setlength{\unitlength}{1mm}\begin{picture}(50,55)\linethickness{0.4mm}\qbezier(45,30)(45,30)(5,30)\qbezier(25.000,10.000)(33.284,10.000)(39.142,15.858)\qbezier(39.142,15.858)(45.000,21.716)(45.000,30.000)\qbezier(45.000,30.000)(45.000,38.284)(39.142,44.142)\qbezier(39.142,44.142)(33.284,50.000)(25.000,50.000)\qbezier(25.000,50.000)(16.716,50.000)(10.858,44.142)\qbezier(10.858,44.142)( 5.000,38.284)( 5.000,30.000)\qbezier( 5.000,30.000)( 5.000,21.716)(10.858,15.858)\qbezier(10.858,15.858)(16.716,10.000)(25.000,10.000)\multiput(5,30)(20,0){3}{\circle*{1}}\put(48,28){\sf (9,-21)}\put(-8,28){\sf (9,9)}\put(21,25){\sf (x,y)}\end{picture}$

Solution :

The coordinates of centre of the circle is equal to the coordinates of mid point of diameter.

Let coordinates of the mid point be (x,y)

Mid point of a line whose end point are (x₁,y₁) and (x₂ ,y₂) is given by ,

$\\ \star \: \boxed{\sf{\purple{(x \:, y) = \bigg( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg) }}}$

Now by comparing the coordinates we have with the formula . We get ,

• x₁ = 9 , x₂ = 9
• y₁ = 9 , y₂ = -21

Now substituting the values ,

$\\ : \implies \sf \: (x,y) = \bigg( \frac{9 + 9}{2} \: ,\frac{9 + ( - 21)}{2} \bigg)$

$\\ : \implies \sf \: (x,y) = \bigg( \frac{18}{2} \: , \frac{12}{2} \bigg)$

$\\ : \implies{\boxed{\pink{\sf{\: (x,y) =( 9 ,\: 6)}}}} \: \bigstar$

Hence , The coordinates of the centre of the circle is (9,6)

Answer 2

Answer:

Given :

Coordinates of end points of diameter are (9,9) and (9,-21)

To Find :

The coordinates of centre of the circle

Figure :

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Solution :

The coordinates of centre of the circle is equal to the coordinates of mid point of diameter.

Let coordinates of the mid point be (x,y)

Mid point of a line whose end point are (x₁,y₁) and (x₂ ,y₂) is given by ,

\begin{gathered} \\ \star \: \boxed{\sf{\purple{(x \:, y) = \bigg( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \bigg) }}} \\ \\ \end{gathered}

⋆

(x,y)=(

2

x

1

+x

2

,

2

y

1

+y

2

)

Now by comparing the coordinates we have with the formula . We get ,

x₁ = 9 , x₂ = 9

y₁ = 9 , y₂ = -21

Now substituting the values ,

\begin{gathered} \\ : \implies \sf \: (x,y) = \bigg( \frac{9 + 9}{2} \: ,\frac{9 + ( - 21)}{2} \bigg) \\ \\ \end{gathered}

:⟹(x,y)=(

2

9+9

,

2

9+(−21)

)

\begin{gathered} \\ : \implies \sf \: (x,y) = \bigg( \frac{18}{2} \: , \frac{12}{2} \bigg) \\ \\ \end{gathered}

:⟹(x,y)=(

2

18

,

2

12

)

\begin{gathered} \\ : \implies{\boxed{\pink{\sf{\: (x,y) =( 9 ,\: 6)}}}} \: \bigstar \\ \\ \end{gathered}

:⟹

(x,y)=(9,6)

★

Hence , The coordinates of the centre of the circle is (9,6)