8.5 Conic sections in polar coordinates  (Page 6/8)

$\frac{x^2}{9}-\frac{y^2}{16}=1$

$\frac{{\left(y-1\right)}^2}{49}-\frac{{\left(x+1\right)}^2}{4}=1$

$x^2-4y^2+6x+32y-91=0$

$2y^2-x^2-12y-6=0$

Center at $\left(0,0\right),$ vertex at $\left(0,4\right),$ focus at $\left(0,\mathrm{-6}\right)$

Foci at $\left(3,7\right)$ and $\left(7,7\right),$ vertex at $\left(6,7\right)$

$y^2=12x$

${\left(x+2\right)}^2=\frac{1}{2}\left(y-1\right)$

$y^2-6y-6x-3=0$

$x^2+10x-y+23=0$

$x^2+4y=0$

${\left(y-1\right)}^2=\frac{1}{2}\left(x+3\right)$

$x^2-8x-10y+46=0$

$2y^2+12y+6x+15=0$

Focus at $\left(\mathrm{-4},0\right);$ directrix is $x=4$

Focus at $\left(2,\frac{9}{8}\right);$ directrix is $y=\frac{7}{8}$

A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

$16x^2+24xy+9y^2+24x-60y-60=0$

$4x^2+14xy+5y^2+18x-6y+30=0$

$4x^2+xy+2y^2+8x-26y+9=0$

$x^2+4xy-2y^2-6=0$

$x^2-xy+y^2-6=0$

$9x^2-24xy+16y^2-80x-60y+100=0$

$x^2-xy+y^2-2=0$

$6x^2+24xy-y^2-12x+26y+11=0$

$r=\frac{10}{1-5\cos \theta }$

$r=\frac{6}{3+2\cos \theta }$

$r=\frac{1}{4+3\sin \theta }$

$r=\frac{3}{5-5\sin \theta }$

$r=\frac{3}{1-\sin \theta }$

$r=\frac{8}{4+3\sin \theta }$

$r=\frac{10}{4+5\cos \theta }$

$r=\frac{9}{3-6\cos \theta }$

Directrix is $x=3$ and eccentricity $e=1$

Directrix is $y=\mathrm{-2}$ and eccentricity $e=4$

$\frac{x^2}{9}+\frac{y^2}{4}=1$

$9y^2+16x^2-36y+32x-92=0$

$\frac{{\left(x-3\right)}^2}{64}+\frac{{\left(y-2\right)}^2}{36}=1$

$2x^2+y^2+8x-6y-7=0$

Write the standard form equation of an ellipse with a center at $\left(1,2\right),$ vertex at $\left(7,2\right),$ and focus at $(4,2).$

A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

$\frac{x^2}{49}-\frac{y^2}{81}=1$

$16y^2-9x^2+128y+112=0$

$\frac{{\left(x-3\right)}^2}{25}-\frac{{\left(y+3\right)}^2}{1}=1$

$y^2-x^2+4y-4x-18=0$

Write the standard form equation of a hyperbola with foci at $\left(1,0\right)$ and $\left(1,6\right),$ and a vertex at $\left(1,2\right).$

$y^2+10x=0$

$3x^2-12x-y+11=0$

${\left(x-1\right)}^2=\mathrm{-4}\left(y+3\right)$

$y^2+8x-8y+40=0$

Write the equation of a parabola with a focus at $\left(2,3\right)$ and directrix $y=\mathrm{-1.}$

A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

$3x^2-2xy+3y^2=4$

$x^2+4xy+4y^2+6x-8y=0$

$11x^2+10\sqrt{3}xy+y^2=4$

$16x^2+24xy+9y^2-125x=0$

$r=\frac{3}{2-\sin \theta }$

$r=\frac{5}{4+6\cos \theta }$

$r=\frac{12}{4-8\sin \theta }$

$r=\frac{2}{4+4\sin \theta }$

Find a polar equation of the conic with focus at the origin, eccentricity of $e=2,$ and directrix: $x=3.$