Circle drawn having it’s diameter equal to focal distance of any point lying on the parabola x² – 4x + 6y + 10 = 0, will touch a fixed line whose equation is -

AB is a double ordinate of the parabola y² = 4ax. Tangents drawn to parabola at A and B meets y-axis at A₁ and B₁ respectively. If the area of trapezium AA₁ B₁ B is equal to 12a², then angle subtended by A₁B₁ at the focus of the parabola is equal to -

The parabola y² = 4x and the circle (x – 6)² + y² = r² will have no common tangent, if ‘r’ is equal to -

The name of the conic represented by the equation x² + y² – 2xy + 20x + 10 = 0 is-

From an external point P tangents are drawn to the parabola y² = 4ax, then the equation to the locus of P when these tangents makes angles θ₁ and θ₂ with the axis, such that tan θ₁ + tan θ₂ with the axis, such that tan θ₁ + tan θ₂ is constant (= b), is -

A point P (t², 2t) lies on the parabola y² = 4x, where FP is produced to B where F is focus. If PB = ƒ(t) and point B always lies on the line y – x = 2, then ƒ(t) is equal to-

Equation of parabola having the extremities of it’s latus rectum as (3, 4) and (4, 3) is -

The set of pointS on the axis of the parabola y² = 4x + 8 from which the three normals to the parabola are all real and different is -

An equilateral triangle SAB is inscribed in the parabola y² = 4ax having it’s focus at ‘S’. If chord AB lies towards the left of S, then side length of this triangle is -

Binding energy per nucleon versas mass number curve for nuclei is shown in the fig W,X,Y and Z are four nuclei indicated on the curve The process that would release energy is

The ends of a line segment are P (1, 3) and Q (1, 1). R is a point on the line segment PQ such that PQ : QR = 1 : λ. If R is an interior point of the parabola y² = 4x, then -

if a point lies inside the curve, then it gives a negative value when we substitute its value in the curve
zero when it lies on the curve and
positive value when it lies outside the curve.

If length of focal chord of y² = 4ax is λ, then angle between axis of parabola and focal chord is

Assertion: If a > 0 and b² – 4ac < 0 then the value of the integral will be of the type tan^–1 + C, where A, B, C,  are constants.
Reason: If a > 0, b² – 4ac < 0 then ax² + b x + c can be written as sum of two squares.

The diagram below shows the propagation of a wave. Which points are in same phase

If f(x) is the primitive of (x  0), then f ' (x) is -

The value of  for which dx = log (4^x + x⁴) is -