Length of Latus Rectum

Q.

Semi latus rectum of the parabola $y^2=4ax$, is the _____ mean between segments of any focal chord of the parabola.

1. Arithmetic

2. Geometric

3. Harmonic

4. None of these

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

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. 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 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 curve for which the normal at any point $P(x,y)$ and the line joining origin to the point $P$ form an isosceles triangle with $x-$axis as base is

1. A rectangular hyperbola
2. An ellipse
3. A parabola
4. A circle

Q.

Semi latus rectum of the parabola $y^2=4ax$, is the _____ mean between segments of any focal chord of the parabola.

1. Arithmetic

2. Geometric

3. Harmonic

4. None of these

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 curve for which the normal at any point $P(x,y)$ and the line joining origin to the point $P$ form an isosceles triangle with $x-$axis as base is

1. An ellipse
2. A parabola
3. A rectangular hyperbola
4. A circle

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

Q. The lengths of the latus rectum of the parabolas
$y^2=12x$ and $x^2=-12y$ are equal.

1. False
2. True

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. 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. Which of the following is incorrect regarding Latus rectum of a rectangular hyperbola?

1. It is perpendicular to $y=x$
2. There can be infinte number of focal chord and latus rectum in a hyperbola.
3. There can be infinite number of focal chord but only two latus rectum in a hyperbola.
4. It passes through focus.