Parabola

The parabola $y=4-x^2$ has vertex P. It intersects $x$-axis at A and B. If the parabola is translated from its initial position to a new position by moving its vertex along the line $y=x+4$, so that it intersects $x$-axis at B and C , then ...

If the locus of mid point of any normal chord of the parabola $y^2=4 x$ is $x-\lambda=\frac{\mu}{y^2}+\frac{y^2}{v}$, where $\lambda, \mu, v \in N$, then ( $\lambda+\mu+v$ ) equals to

The line $y-\sqrt{3} x+3=0$ cuts the parabola $y^2=x+2$ at the points $P$ and $Q$. If the co-ordinates of the point $X$ are $(\sqrt{3}, 0)$, then the value of $X P \cdot X Q$ is

\(\triangle \mathrm{OAB}\) is an equilateral triangle inscribed in the parabola \(\mathrm{y}^2=4 \mathrm{a} x, \mathrm{a}>0\) with O as the vertex, then the length of the side of \(\triangle \mathrm{O A B}\) is

From the focus of the parabola \({y^2} = 12x\), a ray of light is directed in a direction making an angle \({\tan ^{ - 1}}{3 \over 4}\) with x-axis. Then the equation of the line along which the reflected ray leaves the parabola is

Let O be the vertex, 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 the locus of P is :

Let A be the point (0, 4) in the xy-plane and let B be the point (2t, 0). Let L be the midpoint of AB and let the perpendicular bisector of AB meet the y-axis M. Let N be the midpoint of LM. Then locus of N is

If P1P2 and P3P4 are two focal chords of the parabola y2 = 4ax then the chords P1P3 and P2P4 intersect on the

Let the tangent and normal at any point P(at2, 2at), (a > 0), on the parabola y2 = 4ax meet the axis of the parabola at T and G respectively. Then the radius of the circle through P, T and G is

From the point (\(-\)1, \(-\)6), two tangents are drawn to y2 = 4x. Then the angle between the two tangents is

AB is a chord of a parabola y2 = 4ax, (a > 0) with vertex A. BC is drawn perpendicular to AB meeting the axis at C. The projection of BC on the axis of the parabola is

Let P be a point on (2, 0) and Q be a variable point on (y \(-\) 6)2 = 2(x \(-\) 4). Then the locus of mid-point of PQ is

A line passes through the point \(( - 1,1)\) and makes an angle \({\sin ^{ - 1}}\left( {{3 \over 5}} \right)\) in the positive direction of x-axis. If this line meets the curve \({x^2} = 4y - 9\) at A and B, then |AB| is equal to

The point of contact of the tangent to the parabola y2 = 9x which passes through the point (4, 10) and makes an angle \(\theta\) with the positive side of the axis of the parabola where tan\(\theta\) > 2, is

From a point (d, 0) three normal are drawn to the parabola y2 = x, then

The locus of the vertices of the family of parabolas \(6y = 2{a^3}{x^2} + 3{a^2}x - 12a\) is

The equation of the latusrectum of a parabola is x + y = 8 and the equation of the tangent at the vertex is x + y = 12. Then, the length of the latusrectum is

If the line y = x is a tangent to the parabola y = ax2 + bx + c at the point (1, 1) and the curve passes through (\(-\)1, 0), then

The length of the chord of the parabola y2 = 4ax(a > 0) which passes through the vertex and makes an acute angle \(\alpha\) with the axis of the parabola is

Consider the parabola y2 = 4x. Let P and Q be points on the parabola where P(4, \(-\) 4) and Q(9, 6). Let R be a point on the arc of the parabola between P and Q. Then, the area of \(\Delta\)PQR is largest when

Let A, B the two distinct points on the parabola y2 = 4x. If the axis of the parabola touches a circle of radius r having AB as diameter, the slope of the line AB is

Let P(at2, 2at), Q, R(ar2, 2ar) be three points on a parabola y2 = 4ax. If PQ is the focal chord and PK, QR are parallel where the co-ordinates of K is (2a, 0), then the value of r is

Number of common tangents of y = x2 and y = \(-\)x2 + 4x \(-\) 4 is

The area of the region lying above X-axis, and included between the circle x2 + y2 = 2ax and the parabola y2 = ax, a > 0 is

The focus of the conic x2 \(-\) 6x + 4y + 1 = 0 is

The axis of the parabola \({x^2} + 2xy + {y^2} - 5x + 5y - 5 = 0\) is

If the tangent to \({y^2} = 4ax\) at the point \((a{t^2},2at)\) where | t | > 1 is a normal to \({x^2} - {y^2} = {a^2}\) at the point \((a\sec \theta ,a\tan \theta )\), then

If the vertex of the conic \({y^2} - 4y = 4x - 4a\) always lies between the straight lines \(x + y = 3\) and \(2x + 2y - 1 = 0\), then

A line passing through the point of intersection of x + y = 4 and x \(-\) y = 2 makes an angle \({\tan ^{ - 1}}\left( {{3 \over 4}} \right)\) with the X-axis. It intersects the parabola \({y^2} = 4(x - 3)\) at points \(({x_1},{y_1})\) and $...

The locus of the mid-points of all chords of the parabola y2 = 4ax through its vertex is another parabola with directrix

The locus of the middle points of all chords of the parabola y2 = 4ax passing through the vertex is

The coordinates of a moving point P are (2t2 + 4, 4t + 6). Then its locus will be a/an

The vertex of the parabola y2 + 6x \(-\) 2y + 13 = 0 is

If t1 and t2 be the parameters of the end points of a focal chord for the parabola y2 = 4ax, then which one is true?

Find the angle subtended by the double ordinate of length 2a of the parabola y2 = ax at its vertex.

The coordinates of the focus of the parabola described parametrically by x = 5t2 + 2, y = 10t + 4 are

Prove that for all values of m, except zero the st. line \(y = mx + {a \over m}\) touches the parabola y2 = 4ax

If the tangent to the parabola y = x(2 \(-\) x) at the point (1, 1) intersects the parabola at P. Find the co-ordinate of P.

The length of the common chord of the parabolas y2 = x and x2 = y is

If P(at2, 2at) be one end of a focal chord of the parabola y2 = 4ax, then the length of the chord is

The vertex of the parabola x2 + 2y = 8x \(-\) 7 is

The two parabolas x2 = 4y and y2 = 4x meet in two distinct points. One of these is the origin and the other is