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

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Answer 1 · 2018-07-19T16:24:16

Ellipse

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 .

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