Convert the polar equation to rectangular form.

$$r^2=\sin\theta$$

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

$$9x^2+4y^2-90x+8y+228=0$$

Let $(x_1,y_1)$ be the coordinates of a point on the parabola $x^2=4py$. The equation of the line tangent to the parabola at the point is $y-y_1=\frac{x_1}{2p}(x-x_1)$. What is the slope of the tangent line?

Convert the rectangular equation to polar form. Assume $a>0$.

$$y^2=x^3$$

Convert the polar equation to rectangular form.

$$r=8\sin\theta$$

Area Find the dimensions of the rectangle (with sides parallel to the coordinate axes) of maximum area that can be inscribed in the ellipse $x^2/25+y^2/16=1$.

Graphical Reasoning Consider the parabola $x^2=4 p y$

Explain how the result of part (c) can be used as a sketching aid when graphing parabolas.

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Find the greatest (aphelion) and smallest (perihelion) distances from the sun’s center to the comet’s center.

Graphical Reasoning Consider the parabola $x^2=4 p y$

For each parabola in part (a), find the length of the latus recturn (see figure). How can the length of the latus rectum be determined directly from the standard form of the equation of the parabola?

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Use a graphing utility to graph the equation of the orbit.

Graphical Reasoning Consider the parabola $x^2=4 p y$

Locate the focus for each parabola in part (a).

Graphical Reasoning Consider the parabola $x^2=4py$.

Use a graphing utility to graph the parabola for $p=1,p=2,p=3,\ \text{and}, p=4$ and Describe the effect on the graph when increases.

Comet Orbit Halley’s comet has an elliptical orbit, with the sun at one focus. The eccentricity of the orbit is approximately $0.967$. The length of the major axis of the orbit is approximately $35.88$ astronomical units. (An astronomical unit is about $93$ million miles.)

Find an equation of the orbit. Place the center of the orbit at the origin, and place the major axis on the $x$-axis.

Find $dy/dx$ and $d^2y/dx^2$, and find the slope and concavity (if possible) at the given value of the parameter.

$$\underline{Parametric\ Equation}$$

$$x=2t,\ y=3t-1$$

$$\underline{Point}$$

$$t=3$$