Question

If $\left( {x,3} \right)$ and $\left( {3,5} \right)$ are the extremities of a diameter of a circle with center at $\left( {2,y} \right)$ , then find the value of $x$ and $y$
A. $x = 1,y = 4$
B. $x = 4,y = 1$
C. $x = 8,y = 2$
D. None of these

Answer 1

Hint: According to the question it is mentioned that (extremities of a diameter) which means that all the points lie on the same plane, from this statement we can easily apply the midpoint formula to find out the values of x and y.

Formula Used:
Midpoint formula : $\left(\:x,y\:\right)=\:\left[\:\dfrac{x_1\:+\:x_2}{2},\:\:\dfrac{y_1\:+\:y_2}{2}\:\right]$

Complete step by step solution:
In the question it is written that $\left( {x,3} \right)$ and $\left( {3,5} \right)$ are the extremities of a diameter of a circle with center at $\left( {2,y} \right)$which means all the three points lies on the same line where $\left( {2,y} \right)$ will be dividing $\left( {x,3} \right)$ and $\left( {3,5} \right)$ in the ratio $1:1$ .
By using Midpoint formula ,
$\left(\:x,y\:\right)=\:\left[\:\dfrac{x_1\:+\:x_2}{2},\:\:\dfrac{y_1\:+\:y_2}{2}\:\right]\:$
$\left( x \right) = \left[ {\dfrac{{{x_1} + {x_2}}}{2}} \right]$
$\left( y \right) = \left[ {\dfrac{{{y_1} + {y_2}}}{2}} \right]$
$\dfrac{{x + 3}}{2} = 2$ ---- (i)
$\dfrac{{3 + 5}}{2} = y$ ---- (ii)
Simplifying both the equations and finding the value of $x$ and $y$
$\dfrac{{x + 3}}{2} = 2$ ---- (i)
$ \Rightarrow x + 3 = 4$
$\; \Rightarrow x = 1$
Same with equation (ii),
$\dfrac{{3 + 5}}{2} = y$ ---- (ii)
$ \Rightarrow 8 = 2y$
$ \Rightarrow y = 4$
Hence the value of $x$ and $y$ is –
$x = 1,y = 4$ .

Option ‘A’ is correct

Note: Remember the midpoint of the endpoint of the diameter. First, find the midpoint of the line segment made by the endpoint of the diameter. Then compare the midpoint with $\left( {2,y} \right)$ to calculate the value of x and y.