Application of Integrals · Question 4

Let S denote the locus of the mid-points of those chords of the parabola y2=x, such that the area of the region enclosed between the parabola and the chord is 43. Let R denote the region lying in the first quadrant, enclosed by the parabola y2=x, the curve S, and the lines x=1 and x=4. Then which of the following statements is (are) TRUE?

1. (4, 3) S
2. (5, 2) S
3. Area of R is 143-2 3
4. Area of R is 143- 3

Answer

(1) (4, 3) S, (3) Area of R is 143-2 3

Solution

Chord ends (t12,t1),(t22,t2); area between parabola x=y2 and chord = |t1-t2|36= 43 |t1-t2|=2.

Mid-point: k= t1+t22, h= t12+t222= (t1+t2)2+(t1-t2)24=k2+1.

S: x=y2+1. (4, 3): 3+1=4 (true); (5, 2): 3 5.

Area of R= 14 ( x- x-1 )dx= 23(8-1)- 23 3 3= 143-2 3

Answer: (A), (C)