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Mathematics

Assertion (A): The coordinates of one end of a diameter of a circle are (1, 4). If its centre is at (2, -3), then the coordinates of the other end of the diameter are (-3, -10).

Reason (R): The centre of a circle is equidistant from each end of a diameter.

  1. A is true, R is false

  2. A is false, R is true

  3. Both A and R are true

  4. Both A and R are false

Section Formula

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Answer

By mid-point formula,

(x, y) = (x1+x22,y1+y22)\Big(\dfrac{x1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

We know that,

Center is the mid-point of diameter of a circle.

Given,

Center = (2, -3)

One end of diameter = (1, 4)

Let other end of diameter be (a, b).

The coordinates of one end of a diameter of a circle are (1, 4). If its centre is at (2, -3), then the coordinates of the other end of the diameter are (-3, -10). Reflection, RSA Mathematics Solutions ICSE Class 10.

Substituting values, we get :

⇒(2,−3)=(1+a2,4+b2)⇒2=1+a2 and −3=4+b2⇒4=1+a and −6=4+b⇒4−1=a and −6−4=b⇒a=3 and b=−10.\Rightarrow (2, -3) = \Big(\dfrac{1 + a}{2}, \dfrac{4 + b}{2}\Big) \\[1em] \Rightarrow 2 = \dfrac{1 + a}{2} \text{ and } -3 = \dfrac{4 + b}{2} \\[1em] \Rightarrow 4 = 1 + a \text{ and } -6 = 4 + b \\[1em] \Rightarrow 4 - 1 = a \text{ and } -6 - 4 = b \\[1em] \Rightarrow a = 3 \text{ and } b = -10.

The correct coordinates of the other end are (3, -10).

So, Assertion (A) is false.

We know that,

Center is the mid-point of diameter of a circle.

Thus, centre of a circle is equidistant from each end of a diameter.

So, Reason (R) is true.

A is false, R is true.

Hence, Option 2 is the correct option.

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