Parabola

Let L be a tangent line to the parabola y2 = 4x \(-\) 20 at (6, 2). If L is also a tangent to the ellipse \({{{x^2}} \over 2} + {{{y^2}} \over b} = 1\), then the value of b is equal to :

If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a \(\ne\) 0, then 'a' must be greater than :

Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1) is :

If one end of a focal chord AB of the parabola
y2 = 8x is at \(A\left( {{1 \over 2}, - 2} \right)\), then the equation of
the tangent to it at B is :

The locus of a point which divides the line
segment joining the point (0, –1) and a point on
the parabola, x2 = 4y, internally in the ratio
1 : 2, is :

Let a line y = mx (m > 0) intersect the parabola,
y2 = x at a point P, other than the origin. Let
the tangent to it at P meet the x-axis at the point
Q. If area (\(\Delta\)OPQ) = 4 sq. units, then m is equal
to __________.

If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :

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 :

The centre of the circle passing through the
point (0, 1) and touching the parabola y = x2 at the point (2, 4) is :

If the common tangent to the parabolas, y2 = 4x and x2 = 4y also touches the circle, x2 + y2 = c2, then c is equal to :

Let P be a point on the parabola, y2
= 12x and
N be the foot of the perpendicular drawn from
P on the axis of the parabola. A line is now
drawn through the mid-point M of PN, parallel
to its axis which meets the parabola at Q. If the
y-int...

If the tangent to the curve, y = ex
at a point
(c, ec) and the normal to the parabola, y2 = 4x
at the point (1, 2) intersect at the same point on
the x-axis, then the value of c is ________ .

Let the latus ractum of the parabola y2
= 4x be
the common chord to the circles C1
and C2
each of them having radius 2\(\sqrt 5\). Then, the
distance between the centres of the circles C1
and C2
is :

The area (in sq. units) of an equilateral triangle
inscribed in the parabola y2 = 8x, with one of
its vertices on the vertex of this parabola, is :

Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the
origin, on the positive x-axis then which of the following points does not lie on it?

If \(\theta\) denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, the |tan \(\theta\)| is equal to :

Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x is :

Let A(4, \(-\) 4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the area of \(\Delta\)ACB is maximum. Then, the area (in sq. units) of \(\Delta\)ACB, is :

If one end of a focal chord of the parabola,
y2 = 16x is at (1, 4), then the length of this focal
chord is :

The area (in sq. units) of the smaller of the two
circles that touch the parabola, y2 = 4x at the point
(1, 2) and the x-axis is :-

The shortest distance between the line y = x and
the curve y2 = x – 2 is :

The tangent to the parabola y2
= 4x at the point
where it intersects the circle x2
+ y2
= 5 in the
first quadrant, passes through the point :

Let P(4, –4) and Q(9, 6) be two points on the parabola, y2 = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this parabola, such that the area of \(\Delta\)PXQ is maximum. Then this maximum area (in sq. ...

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the rectangle lies inside the parabola, is :

The equation of a tangent to the parabola, x2
= 8y, which makes an angle \(\theta\) with the positive directions of x-axis, is :

Let P be the point of intersection of the common tangents to the parabola y2
= 12x and the hyperbola
8x2
– y2
= 8. If S and S' denote the foci of the hyperbola where S lies on the positive x-axis then P divides SS'
in a ratio :

The tangents to the curve y = (x – 2)2 – 1 at its points of intersection with the line x – y = 3, intersect at the point :

The equation of common tangent to the curves y2
= 16x and xy = –4, is :

If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x – a2) = 0 and the othertwo vertices are the points of intersection of the parabola and y-axis, is 250 sq. units, then a value of 'a' is :

If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c)?

The length of the chord of the parabola x2 \(=\) 4y having equation x – \(\sqrt 2 y + 4\sqrt 2 = 0\)  is -

If the line ax + y = c, touches both the curves x2
+ y2
= 1 and y2
= 4\(\sqrt 2\)x , then |c| is equal to :

Let P be a point on the parabola, x2 = 4y. If the distance of P from the center of the circle, x2 + y2 + 6x + 8 = 0 is minimum, then the equation of the tangent to the parabola at P, is :

Two parabolas with a common vertex and with axes along x-axis and \(y\)-axis, respectively intersect each other in the first quadrant. If the length of the latus rectum of each parabola is \(3\), then the equation of the common tangent to t...

Tangents drawn from the point (\(-\)8, 0) to the parabola y2 = 8x touch the parabola at \(P\) and \(Q.\) If F is the focus of the parabola, then the area of the triangle PFQ (in sq. units) is equal to :

Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the
parabola at A and B, respectively. If C is the centre of the circle through the points P, A and B and \(\angle\)CPB =
\(\theta\), then a v...

If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then \(\left| c \right|\) is equal to :

If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance of P from the origin, is :

P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the normal at P passes through Q, then the minimum value of \(t_1^2\) is :

Let \(P\) be the point on the parabola, \({{y^2} = 8x}\) which is at a minimum distance from the centre \(C\) of the circle, \({x^2} + {\left( {y + 6} \right)^2} = 1\). Then the equation of the circle, passing through \(C\) and having its c...

Let \(O\) be the vertex and \(Q\) be any point on the parabola, \({{x^2} = 8y}\). If the point \(P\) divides the line segment \(OQ\) internally in the ratio \(1:3\), then locus of \(P\) is :

The slope of the line touching both the parabolas \({y^2} = 4x\) and \({x^2} = - 32y\) is

Given : A circle, \(2{x^2} + 2{y^2} = 5\) and a parabola, \({y^2} = 4\sqrt 5 x\).
Statement-1 : An equation of a common tangent to these curves is \(y = x + \sqrt 5\).
Statement-2 : If the line, $$y = mx + {{\sqrt 5 } \over m}\left( {m \n...

If two tangents drawn from a point \(P\) to the parabola \({y^2} = 4x\) are at right angles, then the locus of \(P\) is

A parabola has the origin as its focus and the line \(x=2\) as the directrix. Then the vertex of the parabola is at :

The equation of a tangent to the parabola \({y^2} = 8x\) is \(y=x+2\). The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is :

The locus of the vertices of the family of parabolas
\(y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a\) is :

Let \(P\) be the point \((1, 0)\) and \(Q\) a point on the parabola \({y^2} = 8x\). The locus of mid point of \(PQ\) is :

If \(a \ne 0\) and the line \(2bx+3cy+4d=0\) passes through the points of intersection of the parabolas \({y^2} = 4ax\) and \({x^2} = 4ay\), then :

The normal at the point\(\left( {bt_1^2,2b{t_1}} \right)\) on a parabola meets the parabola again in the point \(\left( {bt_2^2,2b{t_2}} \right)\), then :