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

The point A(2, 3), B(3, 5), C(7, 7) and D(4, 5) are such that 

Q, R and S are the points on the line joining the points P(a, x) and T(b, y) such that PQ = QR = RS = ST.

The line joining A(bcos α, bsin α) and B(acos β, asin β) is produced to point M(x, y) so that AM : MB = b : a, then 

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

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.