Question

Show that the circle $ {{x}^{2}}+{{y}^{2}}-2ax-2ay+{{a}^{2}}=0 $ touches the axes of x and y.

Answer 1

Hint: We first try to form the given circle in its general form of $ {{\left( x-\alpha \right)}^{2}}+{{\left( y-\beta \right)}^{2}}={{r}^{2}} $ to find the centre and the radius. We put the values for the x and y coordinates as 0 and get only 1 value for the other coordinate. For any other number of solutions, the circle doesn’t touch the axis. We also confirm it by using the formula.

Complete step by step answer:

It’s given that the equation of the circle is $ {{x}^{2}}+{{y}^{2}}-2ax-2ay+{{a}^{2}}=0 $ . We transform it in its general form of $ {{\left( x-\alpha \right)}^{2}}+{{\left( y-\beta \right)}^{2}}={{r}^{2}} $ and get $ {{\left( x+a \right)}^{2}}+{{\left( y+a \right)}^{2}}={{a}^{2}} $ . O is the centre.

Equating with the general equation of circle $ {{\left( x-\alpha \right)}^{2}}+{{\left( y-\beta \right)}^{2}}={{r}^{2}} $ , we get the centre as $ O\equiv \left( a,a \right) $ and the radius as a unit.

We get that if it touches any axes then we can put the for the x and y coordinates as 0 and get only 1 value for the other coordinate. If we get no solutions or two solutions, then we can say it doesn’t touch the axes.

We put $ x=0 $ and get $ {{\left( y-a \right)}^{2}}=0 $ which gives $ y=a $ . The circle touches the Y-axis at point $ B\equiv \left( 0,a \right) $ .

Now if we put $ y=0 $ and get $ {{\left( x-a \right)}^{2}}=0 $ which gives $ x=A $ . The circle touches the X-axis at point $ A\equiv \left( a,0 \right) $ .

The circle $ {{x}^{2}}+{{y}^{2}}-2ax-2ay+{{a}^{2}}=0 $ touches the axes of x and y.

Note:

 We also can use the formula that for general equation of circle $ {{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0 $ , if the equation $ {{g}^{2}}=c $ satisfies then it touches the X-axis and if the equation $ {{f}^{2}}=c $ satisfies then it touches the Y-axis. For touching the both axes the required equation would be $ {{f}^{2}}={{g}^{2}}=c $ . In this case $ {{f}^{2}}={{g}^{2}}=c $ satisfies which means it should touch both axes.