Q. 8 - Calculus

Question

Fill in the blanks:

 (a) If the point (r, θ + π) is on the graph of a polar curve whenever the point (r, θ) is on the graph, then the curve is symmetrical .............

(b) If the point ..................is on the graph of a polar curve whenever the point (r, θ) is on the graph, then the curve is symmetrical with respect to the origin. (There is more than one way to answer this question correctly!) 

(c) If the point (−r, −θ) is on the graph of a polar curve whenever the point (r, θ) is on the graph, then the curve is symmetrical..............

Step-by-Step Solution

Answer

The blanks are filled as:

a) about the origin.

b) $\left(-r,\theta \right)$

c) about the y-axis.

Step 1. Given information

a) The points $\left(r,\theta \right),\left(r,\theta +\mathrm{\pi }\right)$ lie on the graph in polar coordinate.

b) The graph is symmetrical about the origin and point $\left(r,\theta \right)$ lies on the graph.

c) The points $\left(r,\theta \right),\left(-r,-\theta \right)$ lies on the graph.

Part (a).Step 1. To find axis of symmetry.

If two points $\left(r,\theta \right)$ and  $\left(r,\theta +\mathrm{\pi }\right)$ lies on the graph of a polar curve then the graph of the curve is symmetric about the origin.

Part (b).Step 1. To find points.

If the graph of a polar curve is symmetric about the origin and point $\left(r,\theta \right)$ lies on the graph then point $\left(-r,\theta \right)$ must lies on the graph.

Part (c).Step 1. To find axis of symmetry.

If points $\left(r,\theta \right)$ and $\left(-r,-\theta \right)$ lies on the graph of a polar curve then the graph of the curve is symmetric about the y-axis.