A variable line `L` intersects the parabola `y=x^(2)` at points `P` and `Q` whose `x`- coordinate are `alpha` and `beta` respectively with `alpha lt beta` the area of the figure enclosed by the segment `PQ` and the parabola is always equal to `4/3`. The variable segment `PQ` has its middle point as `M`
Which of the following is/are correct?

Options

A. `(beta-alpha)` can have more than one real values
B. `(beta-alpha)` can be equal to 2
C. `(beta-alpha)` can have exactly one real value
D. `alpha=2+beta`

Answer

B

Explanation

Any two point on `y=x^(2)` is `P(alpha,alpha^(2)),Q(beta,beta^(2))`
Equation of `PQ, y-alpha^(2)=(alpha+beta)(x-alpha)`
`y=(alpha+beta)x-alpha beta`
Required area `int_(alpha)^(beta)((alpha+beta)x-alpha beta-x^(2))dx`
`implies beta-alpha=2`
Pair of tangents from origin are `y=2x` and `y=-2x`
Area `int_(0)^(1)((x^(2)+)-2x)dx=2/3`