Question

The equation $a{{x}^{2}}+b{{y}^{2}}+2hxy+2gx+2fy+c=0$ represents a circle only if
( a ) A = b , h = 0
( b ) A = b $\ne $0 , h = 0
( c ) A = b $\ne $0, h = 0, ${{g}^{2}}+{{f}^{2}}-c>0$
( d ) A = b $\ne $0, h = 0, ${{g}^{2}}+{{f}^{2}}-ac>0$

Answer 1

Hint: We are given an equation and we have to find out the condition with which this equation forms a circle. For this we are reminded of the standard form of the circle and on the basis of that we find out the condition with which this equation forms a circle.

Complete Step by step solution:
We are given a equation $a{{x}^{2}}+b{{y}^{2}}+2hxy+2gx+2fy+c=0$
We have to find out the condition with which the above condition forms a circle.
To find out the conditions, first we use the condition ${{g}^{2}}+{{f}^{2}}-c>0$
Now we use the complete square method and infer that the equation of circle with centre (-g,-f) and the radius is r.
( This is the general equation because we can choose approximate values for g or f )
We can bring the equation to the form given in the question easily, a = \[{{g}^{2}}+{{f}^{2}}-{{r}^{2}}\]
Thus, \[{{g}^{2}}+{{f}^{2}}-{{a}^{2}}={{r}^{2}}>0\]
Hence, it represents a circle only if a = b $\ne $0, h = 0 , ${{g}^{2}}+{{f}^{2}}-c>0$
Thus, Option (C) is correct.

Note: To solve these questions, we must know the conditions :-
General equation of a conic section is \[A{{x}^{2}}+Bxy+C{{y}^{2}}+Dx+Ey+F=0\]
Where A,B , C, D, E, F are the constants. If
\[{{B}^{2}}-4ac<0\] that is ellipse or circle
\[{{B}^{2}}-4ac=0\] that is parabola
\[{{B}^{2}}-4ac>0\] that is hyperbola
Remember these equations while solving these types of questions. Students must need deep knowledge to solve these types of questions.