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`.

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 join of (x_1,y_1) and (x_2,y_2) makes on obtuse angle at (x_3,y_3), then prove than (x_3-x_1)(x_3-x_2)+(y_3-y_1)(y_3-y_2)<0

If three point A(x_(1), y_(1)), B(x_(2), y_(2)) and C(x_(3), y_(3)) will be collinear then the area of triangleABC= ____.

If the lines x - y - 1 = 0, 4x + 3y = k and 2x – 3y +1 = 0 are concurrent, then k is

If the circle x^2+y^2=a^2 intersects the hyperbola x y=c^2 at four points P(x_1, y_1),Q(x_2, y_2),R(x_3, y_3), and S(x_4, y_4), then x_1+x_2+x_3+x_4=0 y_1+y_2+y_3+y_4=0 x_1x_2x_3x_4=C^4 y_1y_2y_3y_4=C^4

Prove that |(1,x,x^2),(1,y,y^2),(1,z,z^2)|=(x-y)(y-z)(z-x)

If x_1,x_2,x_3 as well as y_1, y_2, y_3 are in G.P. with same common ratio, then prove that the points (x_1, y_1),(x_2,y_2),a n d(x_3, y_3) are collinear.

If x_1,x_2,x_3 as well as y_1, y_2, y_3 are in G.P. with same common ratio, then prove that the points (x_1, y_1),(x_2,y_2),a n d(x_3, y_3) are collinear.

If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x_1,y_1) and (x_2,y_2), respectively, then x_1=y^2 (b) x_1=y_1 y_1=y_2 (d) x_2=y_1

The value of |(1,1,1),(2x,2y,2z),(3x,3y,3z)| is ____