Two points A(1, 4) & B(3, 0) are given on the ellipse `2x^(2) + y^(2) = 18`. The co-ordinates of a point C on the ellipse such that the area of the triangle ABC is greatest is

The line x+2y=1 cuts the ellipse x^(2)+4y^(2)=1 1 at two distinct points A and B. Point C is on the ellipse such that area of triangle ABC is maximum, then find point C.

alpha and beta are eccentric angles of two points A and B on the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 If P(a cos theta,b sin theta) be any point on the same ellipse such that the area of triangle PAB is maximum then prove that theta=(alpha+beta)/(2)

Let F_(1),F_(2) are the foci of the ellipse 4x^(2)+9y^(2)=36 and P is a point on ellipse such that PF_(1)=2PF_(2), then the area of triangle PF_(1)F_(2) is

Given the ellipse (x ^(2))/(4) +y ^(2) =1, the point on the line x =2, such that the tangents drawn from the point to the ellipse are at 45^(@), is

The straight line (x)/(4)+(y)/(3)=1 intersects the ellipse (x^(2))/(16)+(y^(2))/(9)=1 at two points A and B there is a point P on this ellipse such that the area of Delta PAB is equal to 6(sqrt(2)-1) . Then the number of such point (P) is/are (A) 4 (B) 1 (C) 2 (D) 3

The straight line (x)/(4)+(y)/(3)=1 intersects the ellipse (x^(2))/(16)+(y^(2))/(9)=1 at two points A and B there is a point P on this ellipse such that the area of Delta PAB is equal to 6(sqrt(2)-1) .Then the number of such point (P) is/are

Any tangent at a point p(x,y) to the ellipse (x^(2))/(8)+(y^(2))/(18)=1 meets the co-ordinate axes in the points A and B such that the area of the DeltaOAB is least , then the point P is of the form (m,n) where m+n+10 is

Let A=(4, 0), B=(0, 12) be two points in the plane. The locus of a point C such that the area of triangle ABC is 18 sq. units is -

Co-ordinates of a point P((pi)/(3)) on the ellipse 16x^(2) + 25y^(2) = 400 are