If `(y^2-5y+3)(x ^2+x+1)<2x` for all `x in R ,` then find the interval in which `y` lies.

If the points (x_1, y_1),(x_2,y_2), and (x_3, y_3) are collinear show that (y_2-y_3)/(x_2x_3)+(y_3-y_1)/(x_3x_1)+(y_1-y_2)/(x_1x_2)=0

If the ordered pairs (x^(2) - 3x, y^(2) + 4y ) and (-2,5) are equal , then find x and y .

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

Let (x,y) = 2x y ^(2) + 3x ^(2) y calculate u _(x) (2,1) and u_(y) (1,-2).

Find the square root of the expression (4x^2)/(y^2)+(20x)/(y)+13-(30y)/(x)+(9y^2)/(x^2)- (2x)/(y)+5-(3y)/(x) .

Find y' , y'' and y''' if y= x^3 - 6x^2 - 5x +3.

The equation of the transvers and conjugate axes of a hyperbola are, respectively, x+2y-3=0 and 2x-y+4=0 , and their respective lengths are sqrt(2) and 2sqrt(3)dot The equation of the hyperbola is 2/5(x+2y-3)^2-3/5(2x-y+4)^2=1 2/5(x-y-4)^2-3/5(x+2y-3)^2=1 2(2x-y+4)^2-3(x+2y-3)^2=1 2(x+2y-3)^2-3(2x-y+4)^2=1