Latus Rectum

Q. If a variable line, $3x+4y-\lambda =0$ is such that the two circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-18x-2y+78=0$ are on its opposite sides, then the set of all values of $\lambda$ will lie in the interval:

1. $(23,31)$
2. $(2,17)$
3. $[13,23]$
4. $[12,21]$

Q.

The length of the latus rectum of the parabola $9x^2-6x+36y+19=0$

1. 36

2. 9

3. 6

4. 4

Q.

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

1. $2x+y=5$

2. $x+2y=4$

3. $x+3y=5$

4. $x-y=1$

Q.

Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas

\(\\(i)~y^2=8x\\(ii)~4x^2+y=0\\(iii)~y^2-4y-3x+1=0\\(iv)~y^2-4y+4x=0\\(v)~~y^2+4x+4y-3=0\\(vi)~y^2=8x+8y\\(vii)~4(y-1)^2=-7(x-3)\\(viii)~y^2=5x-4y-9\\(ix)~x^2+y=6x-14\)

Q. If the vertex of a parabola be at origin and directrix be x + 5 = 0, then its latus rectum is

1. 10
2. 20
3. 40
4. 5

Q.

If the two circles ${(x-1)}^2+{(y-3)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect in two distinct points, then

1. $2<r<8$

2. $r<2$

3. $r=2$

4. $r>2$

Q. The co – ordinates of the extremities of the latus rectum of the parabola $5y^2=4x$ are

1. 
2. 
3. 
4. 

Q.

Latus rectum of the parabola $y^2-4y-2x-8=0$ is

1. 2

2. 4

3. 8

4. 1

Q. The point (3, 4) is the focus and 2x – 3y + 5 = 0 is the directrix of a parabola. Its latusrectum is

1. None
2. 
3. 
4. 

Q. A circle has its centre at the vertex of the parabola $x^2=4y$ and the circle cuts the parabola at the ends of its latus rectum. The equation of the circle is

1. $x^2+y^2=4$
2. $x^2+y^2=5$
3. $x^2+y^2=1$
4. $x^2+y^2=2$

Q. The latus rectum of a parabola whose directrix is x + y – 2 = 0 and focus is (3, -4), is

1. $-3\sqrt{2}$
2. $3\sqrt{2}$
3. $-3\sqrt{2}$
4. $\frac{3}{\sqrt{2}}$

Q.

Find the area of the triangle formed by the lines joining the vertex of the parabola $x^2=12y$ to the ends of its latus-rectum.

Q. If the curve $y=|x-3|$ touches the parabola $y^2=\lambda (x-4),\lambda >0,$ then length of latus rectum of the parabola(in units) is

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

Q.

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding / cutting machine and a sprayer. It takes 2 h on grinding / cutting machine and 3 h on the sprayer to manufacture a pedestal lamp. It takes 1 h on the grinding / cutting machine and 2 h on the sprayer to Manufacture a shade. On any day, the sprayer is available for at the most 20 h and the grinding / cutting machine for at most 12 h. The profit from the sale of a lamp is Rs. 5 and that from a shades is Rs. 3. Assuming that the manufacture can sell all the lamps and shades that he produce, how should he schedule his daily production in order to maximize his profit?

Q. The three vectors $\hat{i}+\hat{j},\hat{j}+\hat{k},\hat{k}+\hat{i}$ taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume

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

Q. In a triangle OAB, $\angle$AOB = 90º. If P and Q are points of trisection of AB, prove that ${\mathrm{OP}}^2+{\mathrm{OQ}}^2=\frac{5}{9}{\mathrm{AB}}^2$.

Q.

Let and be two unit vectors andθ is the angle between them. Then is a unit vector if

(A) (B) (C) (D)

Q. A circle is drawn through the extremities of the latus rectum of the parabola $y^2=20x$. If the circle touches the directrix of the parabola, then the radius of the circle is(units)

Q. If $2$ and $3$ are the lengths of the segments of any focal chord of a parabola $y^2=4ax$ made by the axis of the parabola, then the length of the latus rectum is

1. $\frac{24}{5}$
2. $\frac{12}{5}$
3. $\frac{11}{5}$
4. $\text{none of these}$

Q. If $\overrightarrow{a,}\overrightarrow{b}$are two non-collinear vectors, prove that the points with position vectors $\overrightarrow{a}+\overrightarrow{b,}\overrightarrow{a}-\overrightarrow{b}\mathrm{and}\overrightarrow{a}+\lambda \overrightarrow{b}$ are collinear for all real values of λ.

Q. All the vertices of a rectangle are of the form $(a,b)$ with $a,b$ integers satisfying the equation $(a–8{)}^2–(b–7{)}^2$ = 5. Then the perimeter of the rectangle is

1. 20
2. 22
3. 24
4. 26

Q.

The co-ordinates of the extremities of the latus rectum of the parabola $5y^2=4x$ are

1. 

2. 

3. 

4. None of these

Q. Consider a parabola with focus $(1,0)$ whose tangent at $(4,2)$ is $y=x-2.$ Then which of the following is (are) CORRECT ?

1. Equation of axis of symmetry is $3x-2y=3$
2. Length of latus rectum is $\frac{2}{\sqrt{13}}$ unit
3. Equation of directrix is $2x+3y=1$
4. Equation of parabola is $9x^2+4y^2-12xy-22x+6y+12=0$

Q.

Show that the points A, B and C with position vectors, , respectively form the vertices of a right angled triangle.

Q.

The ends of latus rectum of parabola $x^2+8y=0$ are

1. (-4, -2) and (4, 2)

2. (4, -2) and (-4, 2)

3. (-4, -2) and (4, -2)

4. (4, 2) and (-4, 2)

Q.

If the vertex of a parabola be at origin and directrix be x+5 = 0 , then its latus rectum is

1. 5

2. 10

3. 20

4. 40

Q. In a quadrilateral ABCD, prove that ${\mathrm{AB}}^2+{\mathrm{BC}}^2+{\mathrm{CD}}^2+{\mathrm{DA}}^2={\mathrm{AC}}^2+{\mathrm{BD}}^2+4{\mathrm{PQ}}^2$, where P and Q are middle points of diagonals AC and BD.

Q. If the parabola $y^2=4ax$ passes through (-3, 2), then length of its latus rectum is

1. 
2. 
3. 
4. 4

Q. The normal a point $(b{t}_1^2,2b{t}_1)$ on a parabola meets the parabola again in the point $(b{t}_2^2,2b{t}_2)$ then

1. 
2. 
3. 
4. 

Q.

Latus rectum of the parabola $y^2-4y-2x-8=0$ is

1. 2

2. 4

3. 8

4. 1