Two Vertices Of A Triangle Are (–1, 4) And (5, 2). If Its Centroid Is (0, –3), Find The Third Vertex.

OPQR is square and M, N are the middle points of the sides PQ and QRrespectively then the ratio of the areas of the square and the triangle OMNis

If p, x₁, x₂….x_i,….and q y₁, y₂,…y_i … are in A.P. with common difference a and b respectively, then locus of the center of mean position of the point A_i (x_i, y_i), i = 1, 2 …n is

If α, β, γ are the real roots of the equation x³ – 3px³ + 3qx – 1 = 0, then the centroid of the triangle with vertices 

The number of points (p, q) such that p, q Ïµ {1, 2, 3, 4} and the equation px² + qx + 1 = 0 has real roots is

If G is the centroid and I the incentre of the triangle with vertices A(–36, 7), B(20, 7) and C(0, –8), then GI is equal to

Consider the point   then

A variable straight line of slope 4 intersects the hyperbola xy = 1 at two points. The locus of the point which divides the line segment between these two points in the ratio 1 : 2 is

Find the co – ordinates of the point which divides the line segment joining the pints (5, – 2) and (9, 6) in the ratio 3 : 1.

Find the co – ordinates of a point which divides externally the line joining (1, –3) and (–3, 9) in the ratio 1 : 3.

Find the area of the pentagon whose vertices are A(1, 1), B(7, 21), C(7, –3), D(12, 2) and (0, –3).