Question

A racing track is build around an elliptical ground whose equation is given by 9x² + 16y² = 144. The width of the track is 3 m as shown below:

Based on given information, answer the following questions:

1. Express y as a function of x from the given equation of ellipse. 1
2. Integrate the function obtained in (i) with respect to x. 1
3. 1. Find the area of the region enclosed within the elliptical excluding the track using integration. 2
      OR
   2. Write the co-ordinates of the points P and Q where the outer edge of the track cuts x axis and y axis in first quadrant and find the area of the triangle formed by points P, O, Q using integration. 2

Answer 1

(i) Given, 9x² + 16y² = 144

Divide both sides by ‘144’

`x^2/16+y^2/9=1`

`x^2/4^2+y^2/3^2=1`

⇒ y = `±3/4sqrt(16-x^2)`

(ii) `intydx=3/4intsqrt16-x^2dx`

`intsqrt(a^2-x^2)dx`

= `x/2sqrta^2-x^2+a^2/2sin^-1x/a+C`

= `3/4[x/2sqrt(16-x^2)+8sin^-1  x/4]+C`

`intydx=(3x)/8sqrt16-x^2+6sin^-1  x/4+C`

(iii) (a) A = `4int_0^4y*dx=4int_0^43/4sqrt(16-x^2)dx`

= `3[x/2sqrt(16-x^2)+8sin^-1  x/4]_0^4`

= `3[(8*pi/2)-(0)]`

= 3 × 4π

= 12 sq.units

(iii) (b) Coordinates of P and Q are (7, 0) and (0, 6) respectively.

`x/7+y/6=1`

⇒ `y/6=1-x/7`

⇒ y = `6-(6x)/7`

Area = `int_0^7(6-(6x)/7)dx`

A = `[6x-(6x^2)/14]_0^7`

= `[6x-3x^2/7]_0^7`

= `[(42-21)-(0)]`

= 21 sq.units