Answer:
Equation 1: r = <u>4</u> +( <u>3</u> * <u>cos theta</u> )
Equation 2: r = sqrt (<u> 5²</u> * <u>sin(2 theta)</u> )
Step-by-step explanation:
GRAPH 1:
The first graph is a dimpled limacon.
General equation for dimpled limacon:
r = a + b cos theta ∴ if dimple is along the x- axis
r = a + b sin theta ∴ if dimple is along the y-axis
y-intercept : { a, -a } = { 4, -4 } ∴ the points at which limacon intersects y-axis
Negative side of x-axis = ( a – b ) ⇒ 1
Positive side of x-axis = ( a + b ) ⇒ 7
Subtract the value of a from sum of a and b to find b:
b = 7 – 4 ⇒ 3
Equation1: r = <u>4</u> +( <u>3</u> * <u>cos theta</u> )
GRAPH 2:
The second graph is a lemniscates.
General equation for lemniscates is:
r² = a² cos(2theta) ∴ if petals of graph are on coordinate axis
r² = a² sin(2 theta) ∴ if petals of graph are not on coordinate axis
now, according to the graph:
a = 5 ⇒ a² = 25
angle of graph: cos2θ, simply divide 360° by 2:
⇒ 180°
The petals cannot be on coordinate axis, we start from 45° and then the next petal will be on:
45° + 180° = 225°
Since the graph is not on the coordinate axis, so
r² = 5² sin(2 theta) ⇒ r = sqrt ( 5² * sin(2 theta) )
Equation 2: r = sqrt (<u> 5²</u> * <u>sin(2 theta)</u> )