Can you solve this??? ^_^
Geometry
Level
2
Find the value of k for which the equation x^2 + y^2 + 4x -2y -k =0 represents a point circle.
The answer is -5.
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1 solution
The general equation of a circle is
( x − H ) 2 + ( y − K ) 2 = R 2
where,
H > abscissa of center of circle
K > ordinate of center of circle
R > radius of circle
Now, the general equation of the circle is
x 2 + y 2 + 2gx + 2fy +c = 0
where,
g= -H
f= -K
c= H 2 + K 2 - R 2
Now, on comparing with the given equation, we get,
H= -2 and K= 1 and c= -k
Now, again, for a point circle, radius should be zero(i.e. R= 0 ).
Therefore, we can apply that
c= H 2 + K 2 - R 2
=> c= ( − 2 ) 2 + ( 1 ) 2 - ( 0 ) 2
=> c= 4 + 1
=> c= 5
=> k = − 5