If `(x+i y)(p+i q)=(x^2+y^2)i` , prove that `x=q ,y=pdot`

If (x+i y)^5=p+i q , then prove that (y+i x)^5=q+i pdot

If p, q are positive integers, f is a function defined for positive numbers and attains only positive values such that f(xf(y))=x^p y^q , then prove that p^2=q .

P N is the ordinate of any point P on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1 and AA ' is its transvers axis. If Q divides A P in the ratio a^2: b^2, then prove that N Q is perpendicular to A^(prime)Pdot

If O is the origin and if the coordinates of any two points Q_1a n dQ_2 are (x_1,y_1)a n d(x_2,y_2), respectively, prove that O Q_1dotO Q_2cos/_Q_1O Q_2=x_1x_2+y_1y_2dot

If x is very large as compare to y , then prove that sqrt(x/(x+y))dotsqrt(x/(x-y))=1+(y^2)/(2x^2) .

If (a-x)/(p x)=(a-y)/(q y)=(a-z)/ra n dp ,q ,a n dr are in A.P., then prove that x ,y ,z are in H.P.

If y_1 and y_2 are the solution of the differential equation (dy)/(dx)+P y=Q , where P and Q are functions of x alone and y_2=y_1z , then prove that z=1+cdote^(-fQ/(y_1)dx), where c is an arbitrary constant.

The LCM and GCD of the two polynomilas is (x^2 + y^(2) ) (x^(4) + x^(2) y^(2) + y^(4)) and x^(2) -y^(2) one of the polynomial q(x) is (x^(4)-y^(4))(x^(2) +y^(2) - xy) find the other polynomials.

If y_1 and y_2 are two solutions to the differential equation (dy)/(dx)+P(x)y=Q(x) . Then prove that y=y_1+c(y_1-y_2) is the general solution to the equation where c is any constant.

A tangent to the ellipse x^2+4y^2=4 meets the ellipse x^2+2y^2=6 at P&Q.Prove that the tangents at P and Q of the ellipse x^2+2y^2=6 are right angle.