If the points (x1, y1),(x2,y2), and (x3...

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`

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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 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

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_3 y_2-y_3 y_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 A(x_1,y_1),B(x_2,y_2) and C(x_3,y_3) are the vertices of traingle ABC and x_(1)^(2)+y_(1)^(2)=x_(2)^(2)+y_(2)^(2)=x_3^(2)+y_(3)^(2) , then show that x_1 sin2A+x_2sin2B+x_3sin2C=y_1sin2A+y_2sin2B+y_3sin 2C=0 .

Normals at (x_(1),y_(1)),(x_(2),y_(2))and(x_(3),y_(3)) to the parabola y^(2)=4x are concurrent at point P. If y_(1)y_(2)+y_(2)y_(3)+y_(3)y_(1)=x_(1)x_(2)x_(3) , then locus of point P is part of parabola, length of whose latus rectum is __________.

If the tangent to the curve x^(3)+y^(3)=a^(3) at the point (x_(1),y_(1)) intersects the curve again at the point (x_(2),y_(2)) , then show that, (x_(2))/(x_(1))+(y_(2))/(y_(1))+1=0 .

If the point (x,y) is equidistant from the points (2,-1) and (-3,4) , then show that , y=x+2 .

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IF x^2+y^2=a^2 , show that, (1+y_1^2)^(3/2)/y_2=-a

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