Question

The circle of radius ‘a’ and center at origin passes through ( \[\sqrt 3 \], 1). What is the value of a?
(a) 2
(b) 4
(c) 3
(d) 1

Answer 1

Hint: The distance between the center and any point on the circle is the length of the radius. The distance between two points (a, b) and (c, d) is given by \[\sqrt {{{(a - c)}^2} + {{(b - d)}^2}} \].

Complete step-by-step answer:
A circle is a two-dimensional figure that has no edges or corners.
The center of the circle is the fixed point whose distance from any points on the circle is equal.
The radius of the circle is the distance between the center of the circle and any point on the circle.
A circle is uniquely represented by its center and its radius. Hence, any circle with a given center and a point on the circle has a particular radius.
The formula to find the distance between two points (a, b) and (c, d) is given as follows:
\[D = \sqrt {{{(a - c)}^2} + {{(b - d)}^2}} ............(1)\]
It is given that the circle has a center point on the origin (0, 0).
The circle also passes through the point ( \[\sqrt 3 \], 1).
Hence, the point ( \[\sqrt 3 \], 1) lies on the circle.
Then the radius of the circle is the distance between (0, 0) and ( \[\sqrt 3 \], 1).
Using equation (1), we have:
\[r = \sqrt {{{(0 - \sqrt 3 )}^2} + {{(0 - 1)}^2}} \]
\[r = \sqrt {{{( - \sqrt 3 )}^2} + {{( - 1)}^2}} \]
Simplifying, we get:
\[r = \sqrt {3 + 1} \]
\[r = \sqrt 4 \]
We know that the value of the square root of 4 is 2.
\[r = 2\text{units}\]
Hence, option (a) is the correct answer.

Note: If you forget the square root in the distance formula, you will end up with the answer that is square of the required answer and choose option (b), which is wrong.