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Answer 1

Question: In a circle, the coordinates of the end points of a diameter are (a,b) and (c,d) then the coordinates of the centre are;

(a) (a/2, b/2) (b) (0, a) (c) (0, b) (d) (a+c/2, b+d/2)

Solution:

We know that the centre of the circle divides the diameter into two equal parts, i.e, the radii.

Hence, we can use the midpoint formula to find out the co-ordinates of the centre.

$\sf Midpoint \ Formula \ \longrightarrow \ \left(\dfrac{x_1 + x_2}{2} , \dfrac{y_1 + y_2}{2}\right)$

Let the Diameter be AB, and the centre be 'O'.

[Refer to the attachment]

A(a,b)

x₁ = a

y1 = b

B(c,d)

x₂ = c

y₂ = d

Using Midpoint formula we get,

$\Longrightarrow \sf O(x,y) = \left(\dfrac{x_1 + x_2}{2} , \dfrac{y_1 + y_2}{2}\right)$

$\Longrightarrow \sf O(x,y) = \left(\dfrac{a + c}{2} , \dfrac{b + d}{2}\right)$

Hence, the answer is Option D.

$\sf Option \ D \longrightarrow \left(\dfrac{a + c}{2} , \dfrac{b + d}{2}\right)$