Parabola Test -...

The combined equation to two parabolas, both have their axis along x-axis, is given by y⁴ - y² (4x + 4 - 2 sin²2α) + sin²2α (4x + 4x + sin² 2α) = 0. The locus of the point of intersection of tangents, one to each of the parabolas, when they include an angle of 90° 
is

The straight line x + y = k + 1 touches the parabola y = x(1 – x) if

Focus and vertex of the parabola that touches x-axis at (1, 0) and x = y at (1, 1) are (h, k) and (p, q) then the value of 25(p + q +h + k)

Number of circles that touch a given parabola and one of its fixed focal chords at focus is

The chord AB of the parabola y² = 4ax cuts the axis of the parabola at C. If   and AC : AB = 1 : 3, then

In a knockout tournament 16 equally skilled players namely P₁, P₂, -------- P₁₆ are participating. In each round players are divided in pairs at random and winner from each pair moves in the next round. If P₂ reaches the semifinal, then the probability that P1 will win the tournament is.

If the parametric equations of the parabola are given by x = 4t² - 2t + 1; y = 3t² + t + 1 and the vertex of the parabola also satisfies y - x = k/100, then the area of the circle x² + y² + 12x -10y + 2k = 0 in square units is 

A parabola has focus at (0, 0) and passes through the points (4, 3) and (–4, –3). The number of lattice points (x, y) on the parabola such that |4x + 3y| < 1000 is

Center of the smallest circle that is drawn to touch the two parabolas given by y² + 2x + 2y + 3 = 0; x² + 2x + 2y + 3 = 0 is

The equation of directrix and latusrectum of a parabola are 3x – 4y + 27 = 0 and 3x – 4y + 2 = 0. Then the length of latusrectum is