Explain how the addition of lactose in the medium regulates the switching on of the \(\textit{lac}\) operon in bacteria.
The lac operon in E. coli is a classic example of gene regulation in bacteria. It controls the expression of genes that code for enzymes involved in the metabolism of lactose. The regulation of the lac operon depends on the presence or absence of lactose (specifically, its isomer allolactose).
Lactose addition leads to the production of allolactose, which binds to the repressor protein, making it inactive and detaching it from the lac operon's operator region. This allows RNA polymerase to transcribe the structural genes, switching on the operon and enabling lactose metabolism.
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following: 
(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates 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 triangle formed by points \(P,O,Q\).