Question

Find the area of the smaller region bounded by the circle with equation `x^2 + y^2=25`, and the lines `y = 1/3x` and `y = -1/3x`?

Answer 1

The area of the smaller region bounded by the circle with equation `x^2 + y^2=25`, and the lines `y = 1/3x` and `y = -1/3x` is `16.088`.

Explanation:

The figure appears as shown in the graph below.

graph{(x^2+y^2-25)(3y-x)(3y+x)=0 [-11.2, 11.2, -5.6, 5.6]}

The area of the smaller region bounded by the circle with equation `x^2 + y^2=25`, and the lines `y = 1/3x` and `y = -1/3x`,

is given by `1/2r^2theta` on one side, where `r` is the radius of circle, here it is `5` and `theta` is the angle between lines `y = 1/3x` and `y = -1/3x` in radians

As slope of the two lines is `1/3` and `-1/3`

the angle between the two lines is given by

`tantheta=(m_1-m_2)/(1+m_1m_2)`

= `(1/3-(-1/3))/(1+(1/3xx-1/3))=(2/3)/(8/9)=2/3xx9/8=3/4`

and `theta=tan^(-1)(3/4)=0.6435` radians. (using scientific calculator)

Hence area is `1/2xx0.6435xx25=8.044`

and on both sides, it is `16.088`