Different curves

Question

I stuck on a following question. The curve is given by: $(3-k)x^{2}+(7-k)y^{2}+9x+9y+7=0$ For which parameter $k$ k the curve will present

1)ellipse or circle 2)parabola 3)hyperbola

Thanks a lot!

Answer 1

$$ax^2+by^2+2gx+2fy+2hxy+c=0$$ represents a general conic .

• If $h=0,a=b$ and $\Delta=0$ then it represent a circle.

• If $a\not=b$,$\Delta=0$ then it represent

  1) Ellipse if $ab-h^2>0$

  2)Parabola if $ab=h^2$

• Hyperbola if $ab-h^2<0$, $\Delta=0$, for rectangular hyperbola $a+b=0$ also.

where $\Delta= \begin{vmatrix}a\ h\ g \\ h\ b\ f\\ g\ f\ c\end{vmatrix}=0$ , which certifies that it's not a pair of straight lines.

Now you have $h=0$ for your case $g=f$ , $a\not=b$ for any $k$ . So it's not a circle. Use $\Delta=0$ to get two possible values of $k$ and see what all curves can it represent.

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If we want only standard curves,then it's easy( with Cartesian axis as axis of the conics)

• $(3-k)(7-k)>0$, represent a ellipse.
• $(3-k)(7-k)<0$ , represent a hyperbola.
• $(3-k)(7-k)=0$ represent a parabola

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