Classifying six conics

Text

For each equation, complete the square, reduce to canonical form and find: type of conic, centre, semi-axes ($a,b$), foci, eccentricity and (for hyperbolas) the asymptotes.

• (a) $4x^2-12x+25y^2-100y+9=0$
• (b) $16x^2-64x+4y^2+4y+29=0$
• (c) $4x^2-20x-64y^2+128y-103=0$
• (d) $-16x^2-32x+9y^2-12y-156=0$
• (e) $2x^2-8x+3y^2+12y+20=0$
• (f) $-4x^2-16x+y^2-2y-15=0$

Solution

(a) $\dfrac{(x-3/2)^2}{25}+\dfrac{(y-2)^2}{4}=1$: ellipse with horizontal major axis, $C\left(\tfrac32;2\right)$, $a=5$, $b=2$, $c=\sqrt{21}$, foci $\left(\tfrac32\pm\sqrt{21};2\right)$, $e=\tfrac{\sqrt{21}}{5}\approx0{,}92$.

(b) $\dfrac{(x-2)^2}{9/4}+\dfrac{(y+1/2)^2}{9}=1$: ellipse with vertical major axis, $C\left(2;-\tfrac12\right)$, $a=\tfrac32$ (horiz.), $b=3$ (vert.), $c=\tfrac{3\sqrt3}{2}$, foci $\left(2;-\tfrac12\pm\tfrac{3\sqrt3}{2}\right)$, $e=\tfrac{\sqrt3}{2}\approx0{,}87$.

(c) $\dfrac{(x-5/2)^2}{16}-\dfrac{(y-1)^2}{1}=1$: hyperbola, horizontal, $C\left(\tfrac52;1\right)$, $a=4$, $b=1$, $c=\sqrt{17}$, foci $\left(\tfrac52\pm\sqrt{17};1\right)$, $e=\tfrac{\sqrt{17}}{4}$, asymptotes $y-1=\pm\tfrac14\left(x-\tfrac52\right)$.

(d) $\dfrac{(y-2/3)^2}{16}-\dfrac{(x+1)^2}{9}=1$: hyperbola, vertical, $C\left(-1;\tfrac23\right)$, $a=4$ (vert.), $b=3$, $c=5$, foci $\left(-1;\tfrac23\pm5\right)$, $e=\tfrac54$, asymptotes $y-\tfrac23=\pm\tfrac43(x+1)$.

(e) $2(x-2)^2+3(y+2)^2=0$: degenerate conic, the single point $(2;-2)$.

(f) $(y-1)^2=4(x+2)^2$: pair of intersecting lines $y=2x+5$ and $y=-2x-3$, with centre $(-2;1)$.

$\boxed{\text{ellipses (a,b); hyperbolas (c,d); point (e); intersecting lines (f)}}$