1.

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?

Answer»

<P>equations of the pair of TANGENTS, drawn to the curve, represented by locus of `M` from origin are `y=2x` and `y=-2x`
equation of pair of tangents to be curve, represented by locus of `M` from origin are `y=x` and `y=-x`
area of the region enclosed between the curve, represented by locus of `M`, and the pair of tangents drawn to it from origin is `2/3` sq. units
area of the region enclosed beween the curve, represented by locus of `M`, and the pair of tangents drawn of it, from origin is `1/3` sq. units

Solution :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`


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