For a > 0 , let the curves C 1 : y 2 = a x and C 2 : x 2 = a y intersect at origin O and a point P . Let the line x = b 0 < b < a intersect the chord O P and the x -axis at points Q and R , respectively. If the line x = b bisects the area bounded by the curves, C 1 and C 2 , and the…

Question

For a &#62; 0 , let the curves C 1 : y 2 = a x and C 2 : x 2 = a y intersect at origin O and a point P . Let the line x = b 0 &#60; b &#60; a intersect the chord O P and the x -axis at points Q and R , respectively. If the line x = b bisects the area bounded by the curves, C 1 and C 2 , and the area of &#8710; O Q R = 1 2 , then &lsquo; a &rsquo; satisfies the equation:

Aicharya · Accepted Answer

&#8747; 0 b a x - x 2 a d x = a 2 6 &#8658; &#8201; 2 3 a b 3 2 - b 3 3 a = a 2 6 &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; &#160; . . . 1 Also area of &#916; O Q R = 1 2 1 2 b 2 = 1 2 &#8658; b = 1 Put in 1 &#8658; &#8201; 4 a a - 2 = a 3 &#8658; &#8201; a 6 + 4 a 3 + 4 = 16 a 3 &#8658; &#8201; a 6 - 12 a 3 + 4 = 0

For a > 0 , let the curves C 1 : y 2 = a x and C 2 : x 2 = a y intersect at origin O and a point P . Let…

Solution

For a &#62; 0 , let the curves C 1 : y 2 = a x and C 2 : x 2 = a y intersect at origin O and a point P . Let…

Solution

For a > 0 , let the curves C 1 : y 2 = a x and C 2 : x 2 = a y intersect at origin O and a point P . Let…