Given The Family Of Lines a(2x + y + 4) + b(x – 2y – 3) = 0. The Number Of Lines Belonging To The Family At A Distance  from Any Point (2, –3) Is

If x₁, x₂, x₃ as well as y₁, y₂, y₃ are in G.P. with the same common ratio, then the points (x₁, y₁), (x₂, y₂) and (x₃, y₃)

Let O(0, 0), P(3, 4), Q(6, 0) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

If a, b, c are all unequal and different from one and the points  are collinear then ab + bc + ca =

Consider three points

              ,

               and

              , then

The lines p(p² + 1)x – y + q = 0 and (p² + 1)² x + (p² + 1)y + 2q = 0 are perpendicular to a common line for

The number of integral values of m, for which the x-coordinates of the point of intersection of the lines 2x + 4y = 9 and 

y = mx + 1 is also an integral is

Let PS be the median of the triangle with vertices P(2, 2), Q(6, –1) andR(7, 3). The equation of the line passing through

(1, –1) and parallel toPS is

Let PQR be a right angled isosceles triangle, right angled at P(2, 1). If the equation of the line QR is 2x + y = 3, then the equation representing the pair of lines PQ and PR is

The point (2, 1) is shifted through a distance  units measured parallel to the line x + y = 1 in the direction of decreasing ordinates to reach Q.The image of Q w.r.t. given line is

Given four lines with equations x + 2y – 3 = 0, 3x + 4y – 7 = 0,

2x + 3y – 4 = 0, 4x + 5y – 6 = 0, then