Straight Lines

The set of all possible values of θ in the interval (0,π) for which the points (1,2) and (sinθ,cosθ) lie on the same side of the line x+y=1 is

The set of all possible values of θ in the interval (0,π) for which the points (1,2) and (sinθ,cosθ) lie on the same side of the line x+y=1 is

The number of integral values of k for which the line, 3x+4y=k intersects the circle, x²+y²-2x-4y+4=0 at two distinct points is.

Let P(h,k) be a point on the curve y = x² + 7x + 2, nearest to the line, y = 3x - 3. Then the equation of the normal to the curve at P is

Two families with three members each and one family with four members are to be seated in a row. In how many ways can they be seated so that the same family members are not separated?

Let L₁ be a tangent to the parabola y²=4(x+1) and L₂ be a tangent to the parabola y²=8(x+2) such that L₁ and L₂ intersect at right angles. Then L₁ and L₂ meet on the straight line: