The tangent and the normal drawn to the curve `y=x^(2)=x+4` at `P(1,4)` cut the X=-axis at A and B respectively. If the length of the subtangent drawn to the curve at `P` is equal to the length of the subnormal, then the area of the triangle `PAB` (in sq units) is

Options

A. 4
B. 32
C. 8
D. 16

Answer

D

Explanation

Given equation of curve is `y-=x^(2)-x+4`,
`implies (dy)/(dx)=2x-1`
Slope of tangent at `P(1,4)` is
`((dy)/(dx))_((1,4))=1`
`:.` Equation of tangent is
`y-4=1(x-1)impliesy-x=3`
and equation of normal at point `P(1,4)` is
`y-4=-1(x-1)impliesx+y=5`
Since the tangent cuts X-axis at `A(-3,0)`
and the normal cuts X-axis at `B(5,0)`
`:.` Area of `DeltaPAB=1/2|(1,4,1),(-3,0,1),(5,0,1)|`
`=1/2|[-4(-3-5)]|=16` sq units