Let S denote the locus of the mid-points of those chords of the parabola y 2 = x , such that the area of the region enclosed between the parabola and the chord is 3 4 . Let R denote the region lying

Question

Let S denote the locus of the mid-points of those chords of the parabola y 2 = x , such that the area of the region enclosed between the parabola and the chord is 3 4 ​ . Let R denote the region lying in the first quadrant, enclosed by the parabola y 2 = x , the curve S , and the lines x = 1 and x = 4 . Then which of the following statements is (are) TRUE?

TestHub · Accepted Answer

​ T = S 1 ​ k y − ( 2 x + h ​ ) = k 2 − h x − 2 k y + 2 k 2 − h = 0 k 2 − h < 0 ⇒ h − k 2 > 0 For Area : interchange x & y y − 2 k x + 2 k 2 − h = 0& y = x 2 x 2 − 2 k x + ( 2 k 2 − h ) = 0 < β α ​ ∣ α − β ∣ = 4 k 2 − 4 ( 2 k 2 − h ) ​ = 4 h − 4 k 2 ​ A = ∫ α β ​ ( ( 2 k x + h − 2 k 2 ) − x 2 ) d x A = 6 ( 4 h − 4 k 2 ) 3/2 ​ = 3 4 ​ ( 4 h − 4 k 2 ) 3/2 = 8 ⇒ ( 4 h − 4 k 2 ) = 4 ​ h − k 2 = 1 ( 4 , 3 ​ ) ∈ S ​ A = ∫ 1 4 ​ ( x ​ − x − 1 ​ ) dx = 3 2 ​ ( x 3/2 − ( x − 1 ) 3/2 ) 1 4 ​ A = 3 2 ​ ( 8 − 3 3 ​ − 1 ) = 3 2 ​ ( 7 − 3 3 ​ ) = 3 14 ​ − 2 3 ​ ​

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