Prove that the greatest value of `x y` is `c^3/sqrt(2a b)`, if `a^2x^4+b^2y^4=c^6dot`

If (alpha,beta) is a point on the circle whose center is on the x-axis and which touches the line x+y=0 at (2,-2), then the greatest value of alpha is (a) 4-sqrt(2) (b) 6 (c) 4+2sqrt(2) (d) +sqrt(2)

If a+2b+3c=4, then find the least value (to the nearest integer) of a^2+b^2+c^2dot

Consider a circle x^2+y^2+a x+b y+c=0 lying completely in the first quadrant. If m_1a n dm_2 are the maximum and minimum values of y/x for all ordered pairs (x ,y) on the circumference of the circle, then the value of (m_1+m_2) is (a) (a^2-4c)/(b^2-4c) (b) (2a b)/(b^2-4c) (c) (2a b)/(4c-b^2) (d) (2a b)/(b^2-4a c)

If a variable tangent to the curve x^2y=c^3 makes intercepts a , b on x a n d y-a x e s , respectively, then the value of a^2b is (a) 27c^3 (b) 4/(27)c^3 (c) (27)/4c^3 (d) 4/9c^3

Prove that the value of each the following determinants is zero: |[(a^x+a^(-x))^2,(a^x-a^(-x))^2 ,1],[(b^y+b^(-y))^2,(b^y-b^(-y))^2 ,1],[(c^z+c^(-z))^2,(c^z-c^(-z))^2, 1]|

Let x ,y ,z ,t be real numbers x^2+y^2=9, z^2+t^2=4 , and xt-y z=6 Then the greatest value of P=xz is a. 2 b. 3 c. 4 d. 6

(a) If x gt a, y gt b and x + y = a + b , then find which one of (sqrtx + sqrty) and (sqrta + sqrtb) is greater ? (b) Which one of the two (sqrt21 + sqrt11) and (2sqrt5 + 2 sqrt3) is greater ? (c) Which one is greater root(3)(4) and root(4)(5) ? (d) Find the greatest and least numbers : 2 sqrt2, 3 root(3)(2), 2 root(4)(3), 2 root(6)(7)

Prove that (a b+x y)(a x+b y)>4a b x y(a , b ,x ,y >0)dot

Suppose a ,b ,c in I such that the greatest common divisor for x^2+a x+b and x^2+bx+c is (x+1) and the least common multiple of x^2+a x+b and x^2+b x+c is (x^3-4x^2+x+6) . Then the value of |a+b+c| is equal to ___________.

Prove that the minimum value of ((a+x)(b+x))/((c+x))a ,b > c ,x > -c is (sqrt(a-c)+sqrt(b-c))^2