Question

Find the radius of the circle whose diameter has endpoints \[( - 3, - 2)\] and $(7,8)$.
A) $5$
B) $5\sqrt 2 $
C) $(2,3)$
D) None of these

Answer 1

Hint: Endpoints of the diameter are given. Using the endpoints given we can find the diameter by the distance formula. Since radius is half of the diameter, dividing by two we get the radius of the circle.

Formula used: Distance formula:
Let \[A({x_1},{y_1})\] and \[B({x_2},{y_2})\] be two points. The distance between $AB$ is given by
$d = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}} $
Radius of a circle is half of its diameter.

Complete step-by-step answer:
Given the points \[( - 3, - 2)\] and $(7,8)$
We have to find the radius of the circle whose diameter has endpoints \[( - 3, - 2)\] and $(7,8)$.
Since the endpoints are given, we can find the length of the line segment using distance formula.
Let \[A({x_1},{y_1})\] and \[B({x_2},{y_2})\] be two points. The distance between $AB$ is given by
$d = \sqrt {{{({x_1} - {x_2})}^2} + {{({y_1} - {y_2})}^2}} $
So here we have the distance between the points \[( - 3, - 2)\] and $(7,8)$
$d = \sqrt {{{( - 3 - 7)}^2} + {{( - 2 - 8)}^2}} $
Simplifying we get,
$d = \sqrt {{{( - 10)}^2} + {{( - 10)}^2}} = \sqrt {{{100}^{}} + {{100}^{}}} $
Again simplifying we get,
$d = \sqrt {200} $
Since, $200 = 100 \times 2$, we have $\sqrt {200} = \sqrt {100 \times 2} = 10\sqrt 2 $
$ \Rightarrow d = 10\sqrt 2 $
Since the points are endpoints of the diameter, here we get,
Diameter of the circle equal to $10\sqrt 2 $.
Now we have, the radius of a circle is half of its diameter.
This gives, radius $ = \dfrac{{10\sqrt 2 }}{2} = 5\sqrt 2 $

$\therefore $ The answer is option B.

Note: A diameter can be drawn for a circle in many ways. So the endpoints of a diameter are not unique. But using each pair of endpoints we can calculate the length of the diameter. Or if we were given the coordinates of the centre and any point on the circle, we can find the radius directly using the distance formula.