Question

The circle ${x^2} + {y^2} + 4x - 4y + 4 = 0{\text{ touches }}$
1. X-axis
2. Y-axis
3. X-axis and Y- axis
4. None of these

Answer 1

Hint: In this question, we start by comparing the given equation of the circle with the standard equation of the circle, from which we will find out the center as well as radius of the circle given. After this, we will simply draw the diagram and find out the required answer.

Complete step by step solution:

The general equation of the circle is

${x^2} + {y^2} + 2gx + 2fy + c = 0$

The given equation of the circle is

${x^2} + {y^2} + 4x - 4y + 4 = 0$

On comparing the above equations, we get

$ \Rightarrow 2gx = 4x \\

\Rightarrow g = 2 \\ $

And

$ \Rightarrow 2fy = - 4y \\

\Rightarrow f = - 2 \\ $

The center of the circle is (-g,-f)

$\therefore C( - 2,2)$

And the radius can be calculated as

$ \Rightarrow r = \sqrt {{g^2} + {f^2} - c} $

$ \Rightarrow r = \sqrt {4 + 4 - 4} \\

\Rightarrow r = \sqrt 4 \\

\Rightarrow r = 2 \\$

$\therefore $ The circle having center (-2,2) and radius r=2 is made below

From the diagram, it is clearly visible that the circle touches X-axis and Y-axis

$\therefore {\text{Option C is correct}}$

Note: We are given with the equation of the circle and the center is at a point other than origin (0,0). Comparing the given equation of the circle with the general form of the equation, we can check where it touches the axis. Also, the condition for the circle to touch the x-axis , y must be zero and to touch y-axis, x must be equal to zero.