Solve: `frac{2}{sqrt x}+frac{3}{sqrt y} = 2`, `frac{4}{sqrt x} -frac{9}{sqrt y} = - 1`, `x > y > 0`

Let a unit vector hat u = x hat i y hat j z hat k make angles (pi)2, (pi)/3 and (2 pi)/3 with the vectors frac{1}{sqrt 2} hat i frac{1}{sqrt 2} hat k , frac{1}{sqrt 2} hat j frac{1}{sqrt 2} hat k and frac{1}{sqrt 2} hat i frac{1}{sqrt 2} hat j respectively. If vec v = frac{1}{sqrt 2} hat i frac{1}{sqrt 2} hat j frac{1}{sqrt 2} hat k , then |hat u - vec v|^2

If the shortest distance between the lines frac{x - lambda} {-2} = frac{y - 2} {1} = frac{z - 1} {1} and frac{x - sqrt 3}{1} = frac{y-1}{-2} = frac{z-2}{1} is 1, then the sum of all possible values of lambda is :

sqrt (2x) -sqrt (3y) = 0sqrt (3x) -sqrt (3y) = 0

{:((2)/(sqrt(x))+ (3)/(sqrt(y)) = 2),((4)/(sqrt(x))-(9)/(sqrt(y)) = -1):}

Solve for x and y : 2/sqrt(x) + 3/sqrt(y) = 2 4/sqrt(x) + 9/sqrt(y) = -1

Solve - frac{sin^(3) frac{x}{2} -cos^(3) frac{x}{2}}{2 +sinx} = cosx/3

Solve dy/dx=frac{3x-4y+2}{4x-5y+3}

If tan A = frac{1}{sqrt (x(x^2 x 1)} , tan B = frac{sqrt x}{sqrt (x^2 x 1)} , and tan C = (x^(-3) x^(-2) x^(-1))^(1/2) , 0 < A, B, C < pi/2 , then A B is equal to :

Prove that yz^(log frac{y}{z}) *(zx)^(log frac{z}{x})* (xy)^(log frac{x}{y}) =1

An ellipse frac{x^2}{a^2}+frac{y^2}{b^2}=1 (a> b) whose eccentircity is frac{1}{sqrt 2} and passes through the focus of the parab whose vertex is (2, 3) and directrix is 2x y - 6 = 0 then the length of the latus rectum of ellipse is