Find the coordinates of a point , where is the diameter of a circle whose centre is and is .

Understand the Question The centre of a circle is always the midpoint of its diameter. For diameter , the centre is the midpoint of and . Using the midpoint formula , we can equate the coordinates to solve for and . Step 1 · Apply Midpoint Formula to Find Coordinates of A Let the coordinates of point be . Given that is the centre and is an endpoint of diameter . Since the centre is the midpoint of diameter : Equating the -coordinate: Equating the -coordinate: Answer > Common Mistakes Midpoint Confusion: Mistakenly finding the midpoint of the centre and point instead of setting the centre as the midpoint of and . Sign Errors in Linear Equations: Adding instead of subtracting when solving , leading to instead of .

Coordinate Geometry | Exercise 7.2

Question 7

Find the coordinates of a point $\text{A}$ , where $\text{AB}$ is the diameter of a circle whose centre is $(2, -3)$ and $\text{B}$ is $(1, 4)$ .

Solution

Understand the Question

The centre of a circle is always the midpoint of its diameter.
For diameter $\text{AB}$ , the centre $\text{C}(2, -3)$ is the midpoint of $\text{A}(x, y)$ and $\text{B}(1, 4)$ .
Using the midpoint formula $\left(\dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2}\right)$ , we can equate the coordinates to solve for $x$ and $y$ .

Step 1 · Apply Midpoint Formula to Find Coordinates of A

Let the coordinates of point $\text{A}$ be $(x, y)$ .

Given that $\text{C}(2, -3)$ is the centre and $\text{B}(1, 4)$ is an endpoint of diameter $\text{AB}$ . Diagram 1

Since the centre $\text{C}$ is the midpoint of diameter $\text{AB}$ : $\left(\dfrac{x + 1}{2}, \dfrac{y + 4}{2}\right) = (2, -3)$

Equating the $x$ -coordinate:

\begin{aligned} 2 &= \dfrac{x + 1}{2} \\[0.6em] 2 \times 2 &= x + 1 \\ 4 &= x + 1 \\ x &= 4 - 1 \\ x &= 3 \end{aligned}

Equating the $y$ -coordinate:

\begin{aligned} -3 &= \dfrac{y + 4}{2} \\[0.6em] -3 \times 2 &= y + 4 \\ -6 &= y + 4 \\ y &= -6 - 4 \\ y &= -10 \end{aligned}

Answer

$(3, -10)$

Common Mistakes

Midpoint Confusion: Mistakenly finding the midpoint of the centre $(2, -3)$ and point $\text{B}(1, 4)$ instead of setting the centre as the midpoint of $\text{A}$ and $\text{B}$ .
Sign Errors in Linear Equations: Adding instead of subtracting when solving $y + 4 = -6$ , leading to $y = -2$ instead of $y = -10$ .

More questions in Exercise 7.2

Find the coordinates of the point which divides the join of $(-1, 7)$ and $(4, -3)$ in the ratio $2 : 3$ .

Find the coordinates of the points of trisection of the line segment joining $(4, -1)$ and $(-2, -3)$ .

To conduct Sports Day activities, in your rectangular shaped school ground $ABCD$ , lines have been drawn with chalk powder at a distance of $1\text{ m}$ each. $100$ flower pots have been placed at a distance of $1\text{ m}$ from each other along $AD$ , as shown in Fig. 7.12. Niharika runs $\dfrac{1}{4}\text{th}$ the distance $AD$ on the 2nd line and posts a green flag. Preet runs $\dfrac{1}{5}\text{th}$ the distance $AD$ on the eighth line and posts a red flag. What is the distance between both the flags? If Rashmi has to post a blue flag exactly halfway between the line segment joining the two flags, where should she post her flag?

Find the ratio in which the line segment joining the points $(-3, 10)$ and $(6, -8)$ is divided by $(-1, 6)$ .

Find the ratio in which the line segment joining $A(1, -5)$ and $B(-4, 5)$ is divided by the $x$ -axis. Also find the coordinates of the point of division.

If $(1, 2)$ , $(4, y)$ , $(x, 6)$ and $(3, 5)$ are the vertices of a parallelogram taken in order, find $x$ and $y$ .

Find the coordinates of a point $\text{A}$ , where $\text{AB}$ is the diameter of a circle whose centre is $(2, -3)$ and $\text{B}$ is $(1, 4)$ .

If A and B are $(-2, -2)$ and $(2, -4)$ , respectively, find the coordinates of P such that $\text{AP} = \dfrac{3}{7} \text{AB}$ and P lies on the line segment AB.

Find the coordinates of the points which divide the line segment joining $A(-2, 2)$ and $B(2, 8)$ into four equal parts.

Q10

Find the area of a rhombus if its vertices are $(3, 0)$ , $(4, 5)$ , $(-1, 4)$ and $(-2, -1)$ taken in order.

[Hint : Area of a rhombus = $\dfrac{1}{2}$ (product of its diagonals)]