Question

The ends of a quadrant of a circle have the co-ordinates (1,3) and (3,1). Then the centre of such of such a circle is
a)(2,2)
b)(1,1)
c)(4,4)
d)(2,6)

Answer 1

Hint: : First we find the distance between 2 end points of the quadrant using the formula$\sqrt{{{\left( {{x}_{1}}-{{x}_{2}} \right)}^{2}}+{{\left( {{y}_{1}}-{{y}_{2}} \right)}^{2}}}$ in which $({{x}_{1}},{{y}_{1}})$and $({{x}_{2}},{{y}_{2}})$ denote the coordinates of those 2 points. Then we find the length of the hypotenuse of the triangle in terms of radius of circle r which is the length of the 2 sides of the triangle and then we equate both the length obtained and find radius. Now we check all the options to find the one which gives the distance between coordinates = radius of the circle.

Complete step-by-step answer:

We are given the question that the points are at the ends of a quadrant.

We can see that the right angled isosceles triangle has radius as both the equal sides so the third side = $\sqrt{2}r$ .


The Distance between both the points can be found as $\sqrt{{{\left( {{x}_{1}}-{{x}_{2}} \right)}^{2}}+{{\left( {{y}_{1}}-{{y}_{2}} \right)}^{2}}}$

Where x and y are the coordinates.

Putting values,


= $\sqrt{{{\left( 1-3 \right)}^{2}}+{{\left( 3-1 \right)}^{2}}}$

= $\sqrt{{{2}^{2}}+{{2}^{2}}}$

= $2\sqrt{2}$

Putting values, we get the distance as 2$\sqrt{2}$


That is equal to the third side which we know is equal to $\sqrt{2}r$.


So, now we equate both equations: 2$\sqrt{2}$= $\sqrt{2}r$


Solving, we find the value of radius = 2 units.


Now we match the coordinates in options and find out the correct radius.


Using the formula to find out the distance between 2 points $\sqrt{{{\left( {{x}_{1}}-{{x}_{2}} \right)}^{2}}+{{\left( {{y}_{1}}-{{y}_{2}} \right)}^{2}}}$.

Let (x, y) be the cp-ordinates of the centre.


R2 = (3 – x)2 + (1-y)2


4 = (3 – x)2 + (1-y)2


Putting (2,2), we get


4 = 2 hence, option a is not the answer.


Putting (1, 1), we get:


4 = 4.


Hence, option b is the correct answer as (1, 1) is the centre of the circle.


Note: While we find the radius of the circle it's necessary to evaluate with precision so that while we put the values in the distance formula, we find the correct radius. Alternatively, by using a graph paper we can draw a circle that has both the points on its circumference and then note down the centre.