If `(x_i ,y_i),i=1,2,3,` are the vertices of an equilateral triangle such that `(x_1+2)^2+(y_1-3)^2=(x_2+2)^2+(y_2-3)^2=(x_3+2)^2+(y_3-3)^2,` then find the value of `(x_1+x_2+x_3)/(y_1+y_2+y_3)` .

If (x+1, y-2)=(3,1), find the values of x and y.

If x^2+y^2=4 then find the maximum value of (x^3+y^3)/(x+y)

If (x+3+1, y-2/3)=(5/3, 1/3) , find the values of x and y.

If log ((x+y)/3)=1/2 (log x +log y) then find the value of x/y+y/x

A triangle has vertices A_(i) (x_(i),y_(i)) for i= 1,2,3,. If the orthocenter of triangle is (0,0) then prove that |{:(x_(2)-x_(3),,y_(2)-y_(3),,y_(1)(y_(2)-y_(3))+x_(1)(x_(2)-x_(3))),(x_(3)-x_(1) ,,y_(3)-y_(1),,y_(2)(y_(3)-y_(1))+x_(2)(x_(3)-x_(1))),( x_(1)-x_(2),,y_(1)-y_(2),,y_(3)(y_(1)-y_(2))+x_(3)(x_(1)-x_(2))):}|=0

If x ,y in R and x^2+y^2+x y=1, then find the minimum value of x^3y+x y^3+4.

Find the following system of equations is consistent, (a+1)^3x+(a+2)^3y=(a+3)^3 (a+1)x+(a+2)y=a+3 x+y=1 , then find the value of a

A triangle has vertices A_i(x_i , y_i)fori=1,2,3 If the orthocentre of triangle is (0,0), then prove that |x_2-x_3y_2-y_3y_1(y_2-y_3)+x_1(x_2-x_3)x_3-x_1y_2-y_3y_2(y_3-y_1)+x_1(x_3-x_1)x_1-x_2y_2-y_3y_3(y_1-y_2)+x_1(x_1-x_2)|=0

If |(3x+4y,-3),(2,x+2y)|=|(7,-3),(2,3)| then x and y are