Show that the graph of the equation r=2 a sinθ, a>0, is a circle of radius a with center at (0,

step 1 Okay, this question asks us about the family of curves equal to R is 2a sine theta. And to get r

Show that the graph of the equation r=2 a sinθ, a>0, is a...

Key Concepts

Polar Coordinates

Polar coordinates are a system for representing points in a plane using a radial distance from a fixed point (called the pole) and an angle measured from a reference direction. This system is especially useful for describing curves that are naturally expressed in terms of distance and direction, rather than horizontal and vertical displacements.

Conversion between Polar and Rectangular Coordinates

This concept involves transforming equations written in polar form (using r and ?) into rectangular form (using x and y) using the relationships x = r cos ? and y = r sin ?. This conversion is crucial for analyzing and graphing curves using the Cartesian coordinate system, and for identifying familiar shapes such as circles and ellipses.

Equation of a Circle in Rectangular Coordinates

In rectangular coordinates, a circle is typically represented by the equation (x ? h)² + (y ? k)² = R², where (h, k) is the center and R is the radius. Recognizing when a converted polar equation matches this form is essential for identifying the geometric properties of the curve, such as its center and radius.

Algebraic Manipulation and Completing the Square

When converting polar equations to rectangular form, algebraic techniques such as squaring both sides and completing the square are often used to rewrite the equation into a familiar standard form. This method is important for isolating terms and revealing the underlying geometry of the curve, particularly in identifying the circle's center and radius.

Transcript

Please give Ace some feedback