Parabola

The parabolas : $a x^2+2 b x+c y=0$ and $d x^2+2 e x+f y=0$ intersect on the line $y=1$. If $a, b, c, d, e, f$ are positive real numbers and $a, b, c$ are in G.P., then :

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2=16 x$. If one of these tangents touches the two curves at $Q$ and $R$, then $(Q R)^2$ is equal to :

A triangle is formed by the tangents at the point (2, 2) on the curves \(y^2=2x\) and \(x^2+y^2=4x\), and the line \(x+y+2=0\). If \(r\) is the radius of its circumcircle, then \(r^2\) is equal to ___________.

If the tangent at a point P on the parabola \(y^2=3x\) is parallel to the line \(x+2y=1\) and the tangents at the points Q and R on the ellipse \(\frac{x^2}{4}+\frac{y^2}{1}=1\) are perpendicular to the line \(x-y=2\), then the area of the ...

The distance of the point \((6,-2\sqrt2)\) from the common tangent \(\mathrm{y=mx+c,m > 0}\), of the curves \(x=2y^2\) and \(x=1+y^2\) is :

The equations of two sides of a variable triangle are \(x=0\) and \(y=3\), and its third side is a tangent to the parabola \(y^2=6x\). The locus of its circumcentre is :

Let a tangent to the curve \(\mathrm{y^2=24x}\) meet the curve \(xy = 2\) at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :

The equations of the sides AB and AC of a triangle ABC are \((\lambda+1)x+\lambda y=4\) and \(\lambda x+(1-\lambda)y+\lambda=0\) respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the p...

If the \(x\)-intercept of a focal chord of the parabola \(y^{2}=8x+4y+4\) is 3, then the length of this chord is equal to ____________.

Let \(\mathrm{PQ}\) be a focal chord of the parabola \(y^{2}=36 x\) of length 100 , making an acute angle with the positive \(x\)-axis. Let the ordinate of \(\mathrm{P}\) be positive and \(\mathrm{M}\) be the point on the line segment PQ su...

Let the tangent to the parabola \(\mathrm{y}^{2}=12 \mathrm{x}\) at the point \((3, \alpha)\) be perpendicular to the line \(2 x+2 y=3\). Then the square of distance of the point \((6,-4)\) from the normal to the hyperbola $$\alpha^{2} x^{2...

Let a common tangent to the curves \({y^2} = 4x\) and \({(x - 4)^2} + {y^2} = 16\) touch the curves at the points P and Q. Then \({(PQ)^2}\) is equal to __________.

Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of \(\Delta\)POQ is equal to ___________.

Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of \({\pi \over 2}\) at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1...

Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of \({\pi \over 4}\) with the line y = 3x + 5 touch the parabola P at A and B. Then the value of a for which A, B and S are collinear is :

Let the focal chord of the parabola \(\mathrm{P}: y^{2}=4 x\) along the line \(\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0\) meet the parabola at the points M and N. Let the line L be a tangent to the hyperbola $$\mathrm{H}: x^{2}-y...

If vertex of a parabola is (2, \(-\)1) and the equation of its directrix is 4x \(-\) 3y = 21, then the length of its latus rectum is :

If the tangents drawn at the points \(\mathrm{P}\) and \(\mathrm{Q}\) on the parabola \(y^{2}=2 x-3\) intersect at the point \(R(0,1)\), then the orthocentre of the triangle \(P Q R\) is :

Two tangent lines \(l_{1}\) and \(l_{2}\) are drawn from the point \((2,0)\) to the parabola \(2 \mathrm{y}^{2}=-x\). If the lines \(l_{1}\) and \(l_{2}\) are also tangent to the circle \((x-5)^{2}+y^{2}=r\), then 17r is equal to __________...

A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola \(y = {\left( {x - {1 \over 4}} \right)^2} + \alpha\), where \(\alpha\) > 0. Then (4\(\alpha\) \(-\) 8)2 is equal to _______...

If the equation of the parabola, whose vertex is at (5, 4) and the directrix is \(3x + y - 29 = 0\), is \({x^2} + a{y^2} + bxy + cx + dy + k = 0\), then \(a + b + c + d + k\) is equal to :

Let \(P(a, b)\) be a point on the parabola \(y^{2}=8 x\) such that the tangent at \(P\) passes through the centre of the circle \(x^{2}+y^{2}-10 x-14 y+65=0\). Let \(A\) be the product of all possible values of \(a\) and \(B\) be the produc...

If the length of the latus rectum of a parabola, whose focus is \((a, a)\) and the tangent at its vertex is \(x+y=a\), is 16, then \(|a|\) is equal to :

Let the common tangents to the curves \(4({x^2} + {y^2}) = 9\) and \({y^2} = 4x\) intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and l ...

Let the normal at the point on the parabola y2 = 6x pass through the point (5, \(-\)8). If the tangent at P to the parabola intersects its directrix at the point Q, then the ordinate of the point Q is :

The equation of a common tangent to the parabolas \(y=x^{2}\) and \(y=-(x-2)^{2}\) is

Let \(\mathrm{P}\) and \(\mathrm{Q}\) be any points on the curves \((x-1)^{2}+(y+1)^{2}=1\) and \(y=x^{2}\), respectively. The distance between \(P\) and \(Q\) is minimum for some value of the abscissa of \(P\) in the interval :

Let \(x = 2t\), \(y = {{{t^2}} \over 3}\) be a conic. Let S be the focus and B be the point on the axis of the conic such that \(SA \bot BA\), where A is any point on the conic. If k is the ordinate of the centroid of the \(\Delta\)SAB, the...

If \(y = {m_1}x + {c_1}\) and \(y = {m_2}x + {c_2}\), \({m_1} \ne {m_2}\) are two common tangents of circle \({x^2} + {y^2} = 2\) and parabola y2 = x, then the value of \(8|{m_1}{m_2}|\) is equal to :

If the line \(y = 4 + kx,\,k > 0\), is the tangent to the parabola \(y = x - {x^2}\) at the point P and V is the vertex of the parabola, then the slope of the line through P and V is :

The sum of diameters of the circles that touch (i) the parabola \(75 x^{2}=64(5 y-3)\) at the point \(\left(\frac{8}{5}, \frac{6}{5}\right)\) and (ii) the \(y\)-axis, is equal to ______________.

The tangents at the points \(A(1,3)\) and \(B(1,-1)\) on the parabola \(y^{2}-2 x-2 y=1\) meet at the point \(P\). Then the area (in unit \({ }^{2}\) ) of the triangle \(P A B\) is :

Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to ___________.

Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x + 2y = ____________.

A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q. If the y-axis bisects the segment PQ, then C is a parabola with :

The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the origin, is :

A tangent line L is drawn at the point (2, \(-\)4) on the parabola y2 = 8x. If the line L is also tangent to the circle x2 + y2 = a, then 'a' is equal to ___________.

A tangent and a normal are drawn at the point P(2, \(-\)4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B respectively. If Q(a, b) is a point such that AQBP is a square, then 2a + b is equal to :

If two tangents drawn from a point P to the parabola y2 = 16(x \(-\) 3) are at right angles, then the locus of point P is :

Let a parabola b be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from O(0, 0) to the parabola P which meet P at S and R, then the area (in sq. uni...

A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it?

A line is a common tangent to the circle (x \(-\) 3)2 + y2 = 9 and the parabola y2 = 4x. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a + c) is equal to _________.

The shortest distance between the line x \(-\) y = 1 and the curve x2 = 2y is :

The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a
moving point of the parabola, is another parabola whose directrix is :

If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x \(-\) 1, then the co-ordinates of P are :

Let y = mx + c, m > 0 be the focal chord of y2 = \(-\) 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4\(\sqrt 2\) (m + c) is equal to _____________.

Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the area (in sq. units) of the triangle PQR is equal to :

If the point on the curve y2 = 6x, nearest to the point \(\left( {3,{3 \over 2}} \right)\) is (\(\alpha\), \(\beta\)), then 2(\(\alpha\) + \(\beta\)) is equal to _____________.

Let P be a variable point on the parabola \(y = 4{x^2} + 1\). Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn from the point P to the line y = x is :

Consider the parabola with vertex \(\left( {{1 \over 2},{3 \over 4}} \right)\) and the directrix \(y = {1 \over 2}\). Let P be the point where the parabola meets the line \(x = - {1 \over 2}\). If the normal to the parabola at P intersects...