How do you convert rectangular equations to polar conics?

How do you convert rectangular equations to polar conics?

Converting a Conic in Polar Form to Rectangular Form r = x 2 + y 2 , x = r c o s θ , and y = r s i n θ .

What is directrix in conics?

The directrix of a conic section is the line which, together with the point known as the focus, serves to define a conic section as the locus of points whose distance from the focus is proportional to the horizontal distance from the directrix, with being the constant of proportionality.

How do you find the directrix of a polar ellipse?

Solution

1. Because sinθ is in the denominator, the directrix is y=p.
2. Since e<1, the conic is an ellipse.
3. Because cosθ is in the denominator, the directrix is x=p.
4. Since e>1, the conic is a hyperbola.
5. Because sine is in the denominator, the directrix is y=−p.
6. Because e=1, the conic is a parabola.

What are sections of a cone?

There are three major sections of a cone or conic sections: parabola, hyperbola, and ellipse(the circle is a special kind of ellipse).

How do you find the polar equation of a conic section?

Polar equations of conic sections: If the directrix is a distance d away, then the polar form of a conic section with eccentricity e is. r ( θ) = e d 1 − e cos. ⁡. ( θ − θ 0), where the constant θ 0 depends on the direction of the directrix. This formula applies to all conic sections.

What is the special case of the formula for conic section?

This formula applies to all conic sections. The only difference between the equation of an ellipse and the equation of a parabola and the equation of a hyperbola is the value of the eccentricity e. There are four important special cases: ( θ). ( θ). ( θ).

How do you rewrite a conic equation in standard form?

Given the polar equation for a conic, identify the type of conic, the directrix, and the eccentricity. Multiply the numerator and denominator by the reciprocal of the constant in the denominator to rewrite the equation in standard form.

What are the three characteristics of a conic section?

Each of these orbits can be modeled by a conic section in the polar coordinate system. Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph. Consider the parabola shown in (Figure).