Question

The general form of unit circle is:
A.${x^2} + {y^2} = {r^2}$
B.${x^2} - {y^2} = {r^2}$
C.${x^2} + {y^2} = 1$
D.${x^2} + {y^2} = 2$

Answer 1

Hint: Write the general form of the equation of the circle with radius $\left( {h,k} \right)$ and centre $r$ is
${\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}$. To find the general form of the unit circle, take the centre as $\left( {0,0} \right)$ and radius as 1 unit. Simplify the equation to write the general form of the unit circle.

Complete step-by-step answer:
A circle is a locus of points that has a fixed distance from a fixed point.
The general form of a circle of centre $\left( {h,k} \right)$ with radius $r$ is ${\left( {x - h} \right)^2} + {\left( {y - k} \right)^2} = {r^2}$
We have to find the general equation for the unit circle, that is the circle with radius as 1 unit.
Let the centre be at origin and radius as 1 unit.
On substituting the value of $\left( {h,k} \right)$ as $\left( {0,0} \right)$ and radius as 1, we get
${\left( {x - 0} \right)^2} + {\left( {y - 0} \right)^2} = {1^2}$
On simplifying the equation we get,
${x^2} + {y^2} = {1^2}$
Hence the general form of the unit circle is ${x^2} + {y^2} = {1}$
Hence, option C is the correct answer.

Note: Since, a circle is a locus of points that has a fixed distance from a fixed point , the equation can be derived using distance formula from a fixed point known as centre and fixed distance is known as radius of the circle. The general form of the circle is also written as, ${x^2} + {y^2} + 2gx + 2fy + c = 0$, where coordinates of the centre is $\left( { - g, - f} \right)$ and the radius is \[\sqrt {{g^2} + {f^2} - c} \].