Question

Find the area of the curve ${x^2} + {y^2} = 2ax$ .
A. $\pi {a^2}$
B. $2\pi {a^2}$
C. $4\pi {a^2}$
D. $\dfrac{1}{2}\pi {a^2}$

Answer 1

Hint: Write the given equation, then add ${a^2}$ to both sides of the equation and form the whole square of $x - a$ . Then write the centre and radius of the circle. Use the formula of area of a circle to obtain the required area.

Formula Used:
${(a - b)^2} = {a^2} - 2ab + {b^2}$
Area of a circle is $\pi {r^2}$ , where r is the radius of the circle.

Complete step by step solution:
The given equation is,
${x^2} + {y^2} = 2ax$
${x^2} + {y^2} - 2ax = 0$
Add ${a^2}$ to both sides of the equation,
 ${x^2} + {y^2} - 2ax + {a^2} = {a^2}$
$\left( {{x^2} - 2ax + {a^2}} \right) + {y^2} = {a^2}$
${(x - a)^2} + {(y - 0)^2} = {a^2}$
This is the equation of a circle with centre $(a,0)$ and radius a.
Hence, the area of the circle is $\pi {a^2}$.

Option ‘A’ is correct

Additional information
Area is the entire amount of space occupied by a flat (2-D) surface or an object's shape.On a sheet of paper, draw a square using a pencil. It has two dimensions. The area of a shape on paper is the area that it occupies.Consider your square as being composed of smaller unit squares. The number of unit squares needed to completely cover the surface area of a certain 2-D shape is used to calculate the area of a figure. Some typical units for measuring area are square cms, square feet, square inches, square meters, etc.

Note: To obtain the area of any specific geometric shape we have some specific formula, to apply that formula we need to first identify the curve. After that we can use the formula, as in this problem first we obtain that the given equation is of a circle, then we have used the area formula of a circle.