Q. 7 - Calculus

Question

Fill in the blanks: 

(a) On a graph that is symmetrical with respect to the xaxis in the polar plane, if the polar point $\left(2,\frac{\mathrm{\pi }}{3}\right)$ is on the graph, then the polar point ............is also on the graph.

(b) On a graph that is symmetrical with respect to the y-axis in the polar plane, if the polar point $\left(2,\frac{\mathrm{\pi }}{3}\right)$ is on the graph, then the polar point ............is also on the graph.

(c) On a graph that is symmetrical with respect to the y-axis in the Cartesian coordinate system, if the point (3, 5) is on the graph, then the point ........is also on the graph. 

(d) On a graph that is symmetrical with respect to the origin in the polar plane, if the polar point  $\left(2,\frac{\mathrm{\pi }}{3}\right)$ is on the graph, then the polar point is also on the graph. 

(e) On a graph that is symmetrical with respect to the origin in the Cartesian coordinate system, if the point (3, 5) is on the graph, then the point.......... is also on the graph. 

Step-by-Step Solution

Answer

The blanks are filled as:

a)$\left(2,-\frac{\mathrm{\pi }}{3}\right)$

b)$\left(-2,-\frac{\mathrm{\pi }}{3}\right)$

c)$\left(-3,5\right)$

d)$\left(-2,\frac{\mathrm{\pi }}{3}\right)$

e)$\left(-3,-5\right)$

Step 1. Given information

Point in polar coordinate is $\left(2,\frac{\mathrm{\pi }}{3}\right)$

Point in the cartesian form is  $\left(3,5\right)$

Part (a). Step 1. Find the point on the graph which is symmetrical about the x-axis.

Graph is symmetrical about the x-axis hence if point $\left(2,\frac{\mathrm{\pi }}{3}\right)$ lies on the graph, then $\left(2,-\frac{\mathrm{\pi }}{3}\right)$ also lies on the graph.

Part (b). Step 1. Find the point on the graph which is symmetrical about the y-axis.

Graph is symmetrical about the y-axis hence if point $\left(2,\frac{\mathrm{\pi }}{3}\right)$ lies on the graph, then $\left(-2,-\frac{\mathrm{\pi }}{3}\right)$ also lies on the graph.

Part (c). Step 1. Find the point on the graph which is symmetrical about the y-axis.

Graph is symmetrical about the y-axis hence if point $\left(3,5\right)$ lies on the graph, then $\left(-3,5\right)$  also lies on the graph.

Part (d). Step 1. Find the point on the graph which is symmetrical about the origin.

The point on thegraph is $\left(2,\frac{\mathrm{\pi }}{3}\right)$

Since graph is symmetrical about the origin so another point $\left(-2,\frac{\mathrm{\pi }}{3}\right)$ also lies on the graph. 

Part (e). Step 1. Find the point on the graph which is symmetrical about the origin.

Since in cartesian plane the point $\left(3,5\right)$ lies on the graph which is symmetrical about the origin., Hence another point $\left(-3,-5\right)$ must lies on the graph.