Question

What conic section does \(r=a \sin \theta+b \cos \theta\) represent? \(?\)

Short answer

The conic section represented by the given equation is a circle with a radius of \(\sqrt{a\^2+b\^2}\).

Step-by-step solution

Identify the shape of the conic section

The given equation is of the form \(r=a \sin \theta + b \cos \theta\). It could represent a circle, ellipse, hyperbola, or parabola in polar coordinates. To identify the shape, square and add the sine and cosine components: \((a \sin \theta)\^2 + (b \cos \theta)\^2 = a\^2 \sin\^2 \theta + b\^2 \cos\^2 \theta\). The trigonometric identity \(\sin\^2x + \cos\^2x = 1\) will come in handy.

Simplify the equation

Substitute the trigonometric identity \(\sin\^2x + \cos\^2x = 1\) into the equation. This gives us \(a\^2 \sin\^2 \theta + b\^2 \cos\^2 \theta = (a\^2 + b\^2)\). Since the right-hand side does not depend on \(\theta\), the curve discriminates a circle whose radius is \(\sqrt{a\^2+b\^2}\). Thus, \(r=a \sin \theta + b \cos \theta\) represents a circle.

Final solution

Conclude that the given equation, \(r=a \sin \theta + b \cos \theta\), represents a circle in polar coordinates with a radius of \(\sqrt{a\^2+b\^2}\).