Distance between Two Parallel Planes

Q.

The equation $x^2-3xy+\lambda y^2+3x-5y+2=0$ when $\lambda$ is a real number, represents a pair of straight lines. If $\theta$ is the angle between the lines, then $\cos e{c}^2\theta$ is equal to

1. $3$

2. $9$

3. $10$

4. $100$

Q.

An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However atleast 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each must be sold in order to maximize the profit for the airline. What is the maximum profit?

Q.

If a vector $\overrightarrow{A}=2\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{B}=3\hat{i}+6\hat{j}+n\hat{k}$ are perpendicular to each other, then the value of $n$ is

Q.

What is the distance between two parallel planes?

Q.

A straight line through the point $A(3,4)$ is such that its intercept between the axes is bisected at $A$. Its equation is

1. $x+y=7$

2. $3x–4y+7=0$

3. $4x+3y=24$

4. $3x+4y=25$

Q.

If the pair of straight lines $xy-x-y+1=0$ and the line $ax+2y-3=0$ are concurrent , then $a=?$

1. $-1$

2. $0$

3. $3$

4. $1$

Q.

A line intersects the $y$-axis and $x$-axis at points $P$ and $Q$ respectively. If the mid point of$PQ$ is $(2,-5)$, then the coordinates of $P$ and $Q$ are:

1. $(0,-5)and(2,0)$

2. $(0,10)and(-4,0)$

3. $(0,4)and(-10,0)$

4. $(0,–10)and(4,0)$

Q.

If the pair of straight lines $x^2-2pxy-y^2=0$and $x^2-2qxy-y^2=0$, be such that each pair bisects the angle between the other pair, then

1. $pq+1=0$

2. $pq–1=0$

3. $p+q=0$

4. $p–q=0$

Q. Distance between two parallel planes 2x+y+2z=8 and 4x+2y+4z+5=0 is

1. $\frac{9}{2}$
2. $\frac{5}{2}$
3. $\frac{7}{2}$
4. $\frac{3}{2}$

Q.

If $\frac{1}{a{b}^{\prime }}+\frac{1}{b{a}^{\prime }}$ = 0, then lines $\frac{x}{b^{\prime }}+\frac{y}{a^{\prime }}$ and $\frac{x}{a}+\frac{y}{b}$ = 1 are ___________________

1. Inclined at 60⁰ to each other

2. Inclined at 30⁰ to each other

3. Parallel lines

4. Perpendicular to each other

Q.

A plane which passes through the point $(3,2,0)$ and the line $\frac{x-3}{1}=\frac{y-7}{5}=\frac{z-4}{4}$ is

1. $x–y+z=1$

2. $x+y+z=5$

3. $x+2y-z=0$

4. $2x-y+z=5$

Q. If the plane $2x-y+2z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4x-2y+4z+\lambda =0$ and $2x-y+2z+\mu =0,$ respectively, then the maximum value of $\lambda +\mu$ is equal to :

1. $5$
2. $9$
3. $13$
4. $15$

Q.

Find the intercepts cut-off by the plane 2x+y-z=5.

Q.

Find distance of point $A\left(2,3\right)$ measured parallel to the line $x-y=5$ from the line $2x+y+6=0$.

Q. The equation of two equal sides $AB$ and $AC$ of isosceles triangle $ABC$ are $x+y=5$ and $7x-y=3$, respectively. Then the equation of the side $BC$ if are of $△ABC=5{\text{unit}}^2$, is/are

1. $3x+y-12=0$
2. $x-3y+21=0$
3. $3x+y-12=0$
4. $x-3y+21=0$

Q. The direction cosines of two lines satisfy the relations $\lambda (l+m)=n$ and $mn+nl+lm=0$.The value of $\lambda$, for which the two lines are perpendicular to each other, is

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

Q.

An equation of a plane parallel to the plane $x-2y+2z-5=0$ and at a unit distance from the origin is

1. $x–2y+2z\pm 3=0$

2. $x–2y+2z+1=0$

3. $x–2y+2z–1=0$

4. $x–2y+2z+5=0$

Q.

Show that the lines $\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}and\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ perpendicular to each other.

Q.

For the lines $2x+5y=7$ and $2x+5y=9,$ which of the following statements is true?

1. Lines are parallel

2. Lines are coincident

3. Lines are intersecting

4. Lines are perpendicular

Q.

Planes are drawn parallel to the coordinate planes through the points

(3, 0, -1) and (-2, 5, 4). Find the lengths of the edges of the parallelopiped so formed.

Q. Distance between parallel planes 2x-2y+z+3=0 and 4x-4y+2z+5=0 is

1. $\frac{2}{3}$
2. $\frac{1}{3}$
3. $\frac{1}{6}$
4. 2

Q.

If the lines $x+3y-9=0,4x+by-2=0$ and $2x-y-4=0$ are concurrent, then $b$ is equal to.

1. $-5$

2. $5$

3. $1$

4. $0$

Q.

Show that the line through the points (1, -1, 2), (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Q. A point $P$ moves in the plane $\prod :2x-3y+6z-4=0$ such that the area of $△PAB$, where $A\equiv (2,2,1)$ and $B\equiv (-1,-4,-1)$ is $14\text{sq. units}$. If the plane perpendicular to the plane $\prod$ containing the locus of $P$ are $6x+ay+bz+d_1=0$ and $6x+ay+bz+d_2=0,d_1>d_2$, then the value of $\left(\frac{d_1-d_2+a+b}{3}\right)$ is

Q.

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0 is
[MP PET 1991]

1. 
   

2. 
   

3. 
   

4. 
   

Q.

Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is

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

2. $\frac{5}{2}$

3. $\frac{7}{2}$

4. $\frac{9}{2}$

Q. If the point $(\alpha ,\alpha )$ lies between the lines $|2x+y|=5$ then select one of the most appropriate option:

1. $|\alpha |<\frac{5}{2}$
2. $|\alpha |<\frac{5}{3}$
3. $|\alpha |<\frac{7}{2}$
4. $|\alpha |<\frac{11}{3}$

Q. Point $(2,3,-5)$, plane $x+2y-2z=9$

Q. Point, Plane: $(3,-2,1),2x-y+2z+3=0$.

Q. Equation of lines
$L_1:2x-2y+3z-2=0=x-y+z+1=0$
$L_2:x+2y-z-3=0=3x-y+2z-1=0$
Distance of point $P(0,0,0)$ from the plane containing $L_1$ and $L_2$ and measured along the line $x=y=z$ is $\frac{a\sqrt{a}}{b}$ (where $a$ and $b$ are coprime numbers) then $a+b$ is