Question

It's either a point, circle, ellipse, or none of above

Answer 1

Consider the given equation,

`4(x-2)^2+5(y+2)^2=0`

Divide the equation by 20 i.e. LCM of 5 and 4,

`((x-2)^2)/(5)+((y+2)^2)/(4)=0\Rightarrow((x-2)^2)/((√(5))^2)+((y+2)^2)/(2^2)=0`

The equation does not resemble an ellipse or circle.

Either it is a point, or none of the above.

Let us check if the equation resembles a point.

Consider the equation as,

`5(y+2)^2=-4(x-2)^2\Rightarrow(y+2)^2=-(4)/(5)(x-2)^2\Rightarrow y+2=\sqrt[]{-(4)/(5)(x-2)^2}`

Simplify the expression further,

`y+2=(x-2).\sqrt[]{(-4)/(5)}\Rightarrow y=2+(x-2).\sqrt[]{(-4)/(5)}`

Now, observe that for every value of 'x' there exist some value of 'y'. It means that the equation corresponds to a point.

But note that there is a negative term in the square root i.e. imaginary term.

It means that this point must lie in a complex plane, not on the real plane.

Therefore, the answer to the given question is a point.nary