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 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 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 tangent at (x_1,y_1) to the curve x^3+y^3=a^3 meets the curve again in (x_2,y_2), then prove that (x_2)/(x_1)+(y_2)/(y_1)=-1

The equation to the chord joining two points (x_1,y_1) and (x_2,y_2) on the rectangular hyperbola xy=c^2 is: (A) x/(x_1+x_2)+y/(y_1+y_2)=1 (B) x/(x_1-x_2)+y/(y_1-y_2)=1 (C) x/(y_1+y_2)+y/(x_1+x_2)=1 (D) x/(y_1-y_2)+y/(x_1-x_2)=1

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

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 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 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 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