Length of Latus Rectum

Q.

The length of the latus-rectum of the parabola $x^2-4x-8y+12=0$ is

1. 4

2. 6

3. 8

4. 10

Q. The equation of the parabola whose focus is at $(-1,-2)$ and the directrix is the line $x-2y+3=0$

1. $4x^2+y^2+4xy+4x+32y+16=0$
2. $4x^2+4y^2+4xy+4x+32y+16=0$
3. $4x^2+y^2+4xy+2x+8y+16=0$
4. $4x^2+y^2+4xy+40x+16y+16=0$

Q. The tangent PT and the normal PN to the parabola $y^2=4ax$ at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola, whose

1. directrix is x = 0
2. vertex is $(\frac{2a}{3},0)$
3. focus is (a, 0)
4. latus rectum is $\frac{2a}{3}$

Q.

The length of the latus-rectum of the parabola $y^2+8x-2y+17=0$ is

1. 2

2. 4

3. 8

4. 16

Q.

The length of the latus-rectum of the parabola $4y^2+2x-20y+17=0$

1. 3

2. 9

3. 6

4. $\frac{1}{2}$

Q. The circle $x^2+y^2+6x-24y+72=0$ and hyperbola $x^2-y^2+6x+16y=46$ intersect at four distinct points. If these four points of intersection lie on a parabola, then

1. the coordinates of the focus of the parabola is $(-3,2)$
2. the coordinates of the focus of the parabola is $(-3,0)$
3. the equation of the directrix of the parabola is $y=0$
4. the length of the latus rectum of the parabola is $4$.

Q. If a parabola passing through point $(-4,-2)$ has its vertex at the origin and $y-$axis as its axis of symmetry, then the length of the latus rectum of the parabola is

Q. If $C$ is a circle described on the focal chord of the parabola $y^2=4x$ as diameter which is inclined at an angle of ${45}^{\circ }$ with the positive $x-$axis, then

1. Radius of the circle is $2$ units
2. The centre of circle is $(3,2)$
3. The line $x+1=0$ touches the circle
4. The circle $x^2+y^2+2x-6y+3=0$ is orthogonal to $C$

Q.

A parabola has the origin as its focus and the line $x=2$as the directrix. Then, the vertex of the parabola is at

1. $(2,0)$

2. $(0,2)$

3. $(1,0)$

4. $(0,1)$

Q. A beam is supported at its ends by two supports which are $12\text{m}$ apart. Since the load is concentrated at its centre, there is a deflection of $3\text{cm}$ at the centre and the deflected beam is in the shape of a parabola. Then distance from the centre where deflection is $1\text{cm},$ is

1. $2\sqrt{6}$
2. $6$
3. $24$
4. $\sqrt{6}$

Q.

If $b$and $c$ are the lengths of the segments of any focal chord of a parabola $y^2=4ax$,

then the length of the latus rectum is$$.

1. $\frac{bc}{\left[b+c\right]}$

2. $\sqrt{bc}$

3. $\frac{[b+c]}{2}$

4. $\frac{2bc}{\left[b+c\right]}$

Q. If the vertex of the parabola is $(2,-3)$ and its directrix is $4x+3y+6=0$, then the length of its latus rectum is

Q. If AFB is a focal chord of the parabola $y^2=4ax$ and AF = 4, FB = 5, then the latus-rectum of the parabola is equal to

1. 
2. 
3. 
4. 

Q.

Locus of all point $P\left(x,y\right)$ satisfy $x^3+y^3+3xy=1$ consist a union of

1. A line and an isolated point

2. A line pair and an isolated point

3. A line and a circle

4. A circle and an isolated point.

Q. If $(2,4)$ and $(2,-4)$ are the end points of latus rectum, then which of the following can be vertex of parabola

1. $(0,0)$
2. $(2,0)$
3. $(4,0)$
4. $(10,0)$

Q. The length of the line segments joining focus to the point of intersection of angular bisector of co-ordinate axes (in the first quadrant) and the parabola $y^2=lx$ is

1. $\frac{l}{4}$
2. $2l$
3. $\frac{5l}{4}$
4. $l\sqrt{2}$

Q. If $(2,0)$ is vertex and $y-$axis is the directrix of a parabola. Then the length of its latus rectum is

1. $2$
2. $4$
3. $8$
4. $16$

Q. The angle made by the latus-rectum of the parabola $y^2=4ax$ at it's vertex is $\theta$ then $3|\tan (\frac{\pi }{4}+\frac{\theta }{2})+\tan (\frac{\pi }{4}-\frac{\theta }{2})|$ is

Q.

Find the coordinates of the point of intersection of the axis and the directrix of the parabola whose focus is (3, 3) and directrix is 3x-4y=2.Find also the length of the latus-rectum.

Q. If $y^2+2y-x+5=0$ represents a parabola, then the length (in units) of the latus rectum is

Q. The total number of focal chord(s) of length $\frac{16}{7}$ in the parabola $7y^2=8x$ is

Q. For the parabola $x^2+7y=0,$ which of the following is/are correct ?

1. Equation of directrix is $4y-7=0$
2. Focus of the parabola is $(0,\frac{7}{4})$
3. Length of latus-rectum is $7$ units.
4. End points of latus-rectum are $(-\frac{7}{2},-\frac{7}{4})$ and $(-\frac{7}{2},\frac{7}{4})$

Q. If $PSQ$ is the focal chord of a parabola such that $SP=2$ and $SQ=4$ then the length of the latus rectum is

1. $\frac{8}{3}$
2. $\frac{16}{3}$
3. $\frac{25}{3}$
4. $\frac{4}{3}$

Q. If the vertex of the parabola is at origin and its directrix is $x-3=0$, then the length of its latus rectum is

Q. The circle $x^2+y^2+6x-24y+72=0$ and hyperbola $x^2-y^2+6x+16y-46=0$ intersect at four distinct points. These four points lie on a parabola, then

1. the focus of parabola is $(–3,2)$
2. the vertex of parabola is $(–3,0)$
3. equation of directrix of parabola is $y=0$
4. length of latus rectum of parabola is $4$
   

Q. The length of the latus rectum of the parabola $3x^2+4y+5+6x=0$ is

1. $\frac{1}{3}$
2. $\frac{2}{3}$
3. $\frac{5}{3}$
4. $\frac{4}{3}$

Q. The length of the latus rectum of the parabola $x=10y^2+by+c,$ where $b$ and $c$ are constants, is $\frac{1}{k}.$ Then $k$ is equal to

Q. A beam is supported at its ends by two supports which are $12\text{m}$ apart. Since the load is concentrated at its centre, there is a deflection of $3\text{cm}$ at the centre and the deflected beam is in the shape of a parabola. Then distance from the centre where deflection is $1\text{cm},$ is

1. $24$
2. $\sqrt{6}$
3. $2\sqrt{6}$
4. $6$

Q.

If b and c are lengths of the segments of any focal chord of the parabola $y^2=4ax,$ then write the lengths of its latus-rectum.

Q. Let PQ be a focal chord of the parabola $y^2=4x$. The tangents to the parabola at P and Q meet at a point on
y = 2x + 1.

Length of chord PQ is

1. 4
2. 3
3. 5
4. 7