If the normals to the ellipse `x^2/a^2+y^2/b^2= 1` at the points `(x_1, y_1), (x_2, y_2) and (x_3, y_3)` are concurrent, prove that `|(x_1,y_1,x_1y_1),(x_2,y_2,x_2y_2),(x_3,y_3,x_3y_3)|=0`.

Three points A(x_1 , y_1), B (x_2, y_2) and C(x, y) are collinear. Prove that: (x-x_1) (y_2 - y_1) = (x_2 - x_1) (y-y_1) .

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 points (x_1,y_1),(x_2,y_2)and(x_3,y_3) are collinear, then the rank of the matrix {:[(x_1,y_1,1),(x_2,y_2,1),(x_3,y_3,1)]:} will always be less than

Write the condition of collinearity of points (x_1,\ y_1),\ \ (x_2,\ y_2) and (x_3,\ y_3) .

If three points (x_1,y_1),(x_2, y_2),(x_3, y_3) lie on the same line, prove 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 join of (x_1,y_1) and (x_2,y_2) makes on obtuse angle at (x_3,y_3), then prove that (x_3-x_1)(x_3-x_2)+(y_3-y_1)(y_3-y_2)<0

Write the formula for the area of the triangle having its vertices at (x_1,\ y_1),\ \ (x_2,\ y_2) and (x_3,\ y_3) .

An equilateral triangle has each side to a. If the coordinates of its vertices are (x_(1), y_(1)), (x_(2), y_(2)) and (x_(3), y_(3)) then the square of the determinat |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)| equals

The lines 2x + y -1=0 , ax+ 3y-3=0 and 3x + 2y -2=0 are concurrent for -

If the coordinates of the vertices of an equilateral triangle with sides of length a are (x_(1),y_(1)),(x_(2),y_(2)) and (x_(3),y_(3)) then |(x_(1),y_(1),1),(x_(2),y_(2),1),(x_(3),y_(3),1)|^2=(3a^(4))/4