How do you use the discriminant to classify the conic section #x^2 + y^2 - 6x + 8y - 24 = 0#?

1 Answer

Answer:

Ellipse

Explanation:

The given general quadratic equation:

#x^2+y^2-6x+8y-24=0#

Comparing above equation with the standard form of quadratic equation: #ax^2+2hxy+by^2+2gx+2fy+c=0# we get

#a=1, h=0, b=1, g=-3, f=4, c=-24#

Now, using determinant #(\Delta)# of quadratic equation as follows

#\Delta=abc+2fgh-af^2-bg^2-ch^2#

#=(1)(1)(-24)+2(4)(-3)(0)-(1)(4)^2-(1)(-3)^2-(-24)(0)^2#

#=-49#

#\because Delta\ne 0# hence the given quadratic equation shows a conic section. (#Delta=0# is the case of pair of lines)

Now, using determinant of conic section :

#h^2-ab#

#=0^2-1\cdot 1#

#=-1#

#\because h^2-ab<0# hence the given quadratic equation shows an ellipse .