If `x/y= (a+2)/(a-2)` show that `(x^2-y^2)/(x^2+y^2)=(4a)/(a^2+4)`

If x/y = (a + 2)/(a - 2) , then show that (x^(2) - y^(2))/(x^(2) + y^(2)) = (4a)/(a^(2) + 4) .

If y= (tan^(-1)x)^(2) , show that (x^(2)+1)^(2) y_(2)+2x(x^(2)+1)y_(1)=2 .

If x+a= a(2 cos theta-cos 2 theta) and y= a(2 sin theta- sin 2 theta) ,show that (x^(2)+y^(2)+2ax)^(2)=4a^(2)(x^(2)+y^(2)) .

If x^(2)+y^(2)=t+(1)/(t) and x^(4)+y^(4)=t^(2)+(1)/(t^(2)) then show that (dy)/(dx)=-(y)/(x) .

If y=log(x+sqrt(x^2+a^2)) , show that (a^2+x^2)y_2+xy_1=0

If x+y=15 , show that x^(2)+y^(2) is least , when x=y=(15)/(2) .

If y^2 (1-x^2) = x^2 +1 , show that (1-x^4) (dy/dx)^2 = y^4 -1 .

If y= 2^((1)/(log_(x)4)) then prove that x=y^(2) .

A circle of constant radius a passes through the origin O and cuts the axes of coordinates at points P and Q . Then the equation of the locus of the foot of perpendicular from O to P Q is (A) (x^2+y^2)(1/(x^2)+1/(y^2))=4a^2 (B) (x^2+y^2)^2(1/(x^2)+1/(y^2))=a^2 (C) (x^2+y^2)^2(1/(x^2)+1/(y^2))=4a^2 (D) (x^2+y^2)(1/(x^2)+1/(y^2))=a^2

If (3x-4y) prop sqrt(xy) then show that (x^2+y^2) prop xy .