[u+2y=-1quad " Byelinaivation "],[alpha u-3y=12]

If U(x,y,z) = x ^(2) + 2xy + y ^(2) z - z ^(2) find the linear approximation for U at (1, -1,2)

if x and y coordinates of a point P in x-y plane are given by x=(u cos alpha)t,y=(u sin alpha)t-(1)/(2)gt^(2) where t is a aprameter and u,alpha,g the constants.Then the locus of the point P is a parabola then whose vertex is:

If y=(u)/(v) , where u and v are functions of x, show that v^(3)(d^(2)y)/(dx^(2))=|{:(u,v,0),(u',v',v),(u'',v'',2v'):}| .

If y=(u)/(v) , where u and v are functions of x, show that v^(3)(d^(2)y)/(dx^(2))=|{:(u,v,0),(u',v',v),(u'',v'',2v'):}| .

If y=(u)/(v) , where u and v are functions of x, show that v^(3)(d^(2)y)/(dx^(2))=|{:(u,v,0),(u',v,v),(u'',v'',2v'):}| .

Given that the vector alpha and beta are non-collinear. The value of x and y for which u-v=w holds true if u= 2 x alpha + y beta, v= 2 y alpha + 3 x beta and w= 2alpha -5 beta , are

Prove that the circles x^2+y^2+24ux+2vy=0 and x^2+y^2+2u_1x+2v_1y=0 touch each other externally if -12u_1u=v_1v .

Prove that the circles x^2+y^2+24ux+2vy=0 and x^2+y^2+2u_1x+2v_1y=0 touch each other externally if -12u_1u=v_1v .

If (x + iy)^3 = u + iv , then show that : u/x+v/y=4 (x^2-y^2) .

If (x+iy)^(3) = u + iv , then show that u/x + v/y = 4(x^(2)-y^(2))