The combined equation to two parabolas, both have their axis along xaxis, is given by y^{4}  y^{2} (4x + 4  2 sin^{2}2α) + sin^{2}2α (4x + 4x + sin^{2} 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
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Focus and vertex of the parabola that touches xaxis 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 y^{2} = 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 P_{1}, P_{2},  P_{16} are participating. In each round players are divided in pairs at random and winner from each pair moves in the next round. If P_{2} reaches the semifinal, then the probability that P1 will win the tournament is.
If the parametric equations of the parabola are given by x = 4t^{2}  2t + 1; y = 3t^{2} + t + 1 and the vertex of the parabola also satisfies y  x = k/100, then the area of the circle x^{2} + y^{2} + 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 y^{2} + 2x + 2y + 3 = 0; x^{2} + 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
Let A ≡ (9, 6), B(4, 4) be two points on parabola y^{2} = 4x and P(t^{2}, 2t), t∈[2, 3] be a variable point on it such that area of DPAB is maximum, then point P will be
Two tangents to the parabola y^{2} = 4x, one drawn at a point P and another drawn at the point where the normal at the image of P in the axis of the parabola meets the curve again, include an angle
If the chord joining the points t_{1} and t_{2} on the parabola y^{2} = 4ax subtends a right angle at its vertex then t_{2} =
The equation of the latusrectum of the parabola y^{2} – 6y + 4x – 3 = 0 is
The equation of the latusrectum of the parabola y^{2} – 6y + 4x – 3 = 0 is
If the line x – 1 = 0 is the directrix of the parabola y^{2} – kx + 8 = 0, then one of the value of k is
The curve described parametrically by x = t^{2} + t + 1, y = t^{2}  t + 1 represents
The length of the latus rectum of the parabola 169{(x  1)^{2} + (y  3)^{2}} = (5x  12y + 17)^{2}} is
The angle between the tangents drawn from the point (3, 4) to the parabola y^{2}  2y + 4x = 0 is
Equation of common tangent of parabola y^{2} = 8x and x^{2} + y = 0 is
If the tangents at two points (1, 2) and (3, 6) as a parabola intersect at the point (– 1, 1), then the slope of the directrix of the parabola is
A variable chord PQ of the parabola y = 4x^{2} substends a right angle at the vertex. Then the locus of points of intersection of the tangents at P and Q is
Point on the curve y^{2} = 4(x 10) which is nearest to the line x + y = 4 may be
If (h, k) is a point on the axis of parabola 2(x 1)^{2} + 2 (y 1)^{2} = (x + y + 2)^{2} from where three distinct normals can be drawn, then
The equation of the common tangent touching the circle (x – 3)^{2} + y^{2} = 9 and the parabola y^{2} = 4x below y^{2}  4x below the x–axis is
The line x + y = 6 is a normal to the parabola y^{2} = 8x at the point
The tangent and normal at the point P(4, 4) to the parabola, y^{2} = 4x intersect the x–axis at the points Q and R respectively. Then the cirucm centre of the ΔPQR is
All chords of the parabola y^{2} = 4x which subtend right angle at the origin are concurrent at the point:
The mirror image of the parabola y^{2} = 4x in the tangent to the parabola to the point (1,2) is
The locus of mid–point of family of chords λx + y  5 = 0 (parameter) of the parabola x^{2} = 20y is
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209 videos443 docs143 tests
