Section Formula

Q.

Find the ratio in which the line segment joining the points (2, 4, 5) and (3, -5, 4) is divided by the yz-plane.

Q.

Find the distance of a point (2, 4, -1) from the line. $\frac{x+5}{1}=\frac{y+3}{4}=\frac{z-6}{-9}$.

Q. The points $(a,0),(0,b)$ and $(1,1)$ are collinear. Then the value of $\frac{1}{a}+\frac{1}{b}$ is

Q.

A rhombus of side $20\mathrm{cm}$ has two angles of ${60}^{\circ }$ each .Find the length of the diagonal.

Q. The line segment joining the points $(3,-4)$ and $(1,2)$ is trisected at the point $P$ and $Q.$ If the coordinates of $P$ and $Q$ are $(p,-2)$ and $(\frac{5}{3},q)$ respectively, then the value of $3p-q$ is

Q.

If $(a,b,c)$is the image of the point $(1,2,-3)$in the line $\frac{x+1}{2}=\frac{y-3}{-2}=\frac{z}{-1}$then $\mathit{a}\mathbf{+}\mathit{b}\mathbf{+}\mathit{c}$ is

1. $2$

2. $3$

3. $-1$

4. $1$

Q.

The ratio in which the line joining (2, 4, 5) and (3, 5, -9) is divided by the

yz-plane is

1. 4 : -3
2. 3 : 2
3. -2 : 3
4. 2 : 3

Q.

An equilateral triangle is inscribed in the parabola $y^2=4ax$, where one vertex is at the vertex of the parabola.

Find the length of the side of the triangle.

1. $8a\sqrt{3}$

2. $4a\sqrt{3}$

3. $2a\sqrt{3}$

4. $a\sqrt{3}$

Q.

Which of the following statments are correct?

1. The coordinates of the point which divides the line segment joining the points

$(1,-2,3)$ and $(3,4,-5)$ internally in the ratio $2:3$ is $(-3,-14,19)$.

2. The coordinates of the point which divides the line segment joining the points

$(1,-2,3)$ and $(3,4,-5)$ externally in the ratio $2:3$ is $(\frac{9}{5},\frac{2}{5},\frac{-1}{5})$.

1. Only 1

2. Only 2

3. Both 1 & 2

4. None of these

Q. The ratio in which yz- plane divides the segment joining the points (-2, 4, 7) and (3, -5, 8) is

1. 2 : 3
2. 4 : 5
3. 3 : 2
4. -7 : 8

Q.

The intercepts of the plane $2x-3y+4z=12$ on the coordinate axes are given by

1. $3,-2,15$

2. $6,-4,3$

3. $6,-4,-3$

4. $2,-3,4$

Q. If $A(4,1),B(7,4),C(13,-2)$ are the three consecutive vertices of a rectangle $ABCD$, then the coordinates of $D$ are

1. $(-10,-5)$
2. $(-10,5)$
3. $(10,5)$
4. $(10,-5)$

Q. The plane which bisects the line joining the points $(4,-2,3)$ and $(2,4,-1)$ at right angles also passes through the point:

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

Q. $A(1,-1,-3),B(2,1,-2),C(-5,2,-6)$ are the position vectors of the vertices of a triangle ABC. The length of the bisector of its internal angle at A is :

1. $\frac{3\sqrt{10}}{2}$
2. $\frac{3\sqrt{10}}{4}$
3. $\frac{3\sqrt{10}}{5}$
4. $\frac{3\sqrt{10}}{7}$

Q.

There are $\mathrm{m}$ points on a straight line $\mathrm{AB}$ and $\mathrm{n}$ points on another line $\mathrm{AC}$, none of them being the point $\mathrm{A}$. Triangles are formed from these points as vertices when

$\left(\mathrm{i}\right)$ $\mathrm{A}$ is excluded

$\left(\mathrm{ii}\right)$ $\mathrm{A}$ is included.

Then the ratio of the number of triangles in the two cases is?

1. $\frac{\mathrm{m}+\mathrm{n}-2}{\mathrm{m}+\mathrm{n}}$

2. $\frac{\mathrm{m}+\mathrm{n}-2}{2}$

3. $\frac{\mathrm{m}+\mathrm{n}-2}{\mathrm{m}+\mathrm{n}+2}$

4. None of these.

Q.

The point of intersection of the line joining the points $(3,4,1)$ and $(5,1,6)$ and the $xy$ plane is

1. $(13,23,0)$

2. $\left(\frac{13}{5},\frac{23}{5},0\right)$

3. $(-13,23,0)$

4. $\left(-\frac{13}{5},\frac{23}{5},0\right)$

Q. If $A(1,1)$ and $B(2,-3)$ are two points and $P(h,k)$ lies on the line $AB$ such that $PA=3AB$, where $B$ lies in between $A$ and $P$, then the value of $h-k$ is

Q. Let the position vectors of points $A$ and $B$ be $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k}$, respectively. A point $p$ divides the line segment $AB$ internally in the ratio $\lambda :1(\lambda >0)$. If $O$ is the origin and $\overrightarrow{OB}\cdot \overrightarrow{OP}-3{|\overrightarrow{OA}\times \overrightarrow{OP}|}^2=6$, then $\lambda$ is equal to

Q. If three points $A,B$ and $C$ lie on a line and $A\equiv (3,4),$ $B\equiv (7,7)$ and $AC=10$, then the coordinates of the point $C$ can be

1. $(10,11)$
2. $(11,10)$
3. $(-5,-2)$
4. $(5,2)$

Q.

Given that P (3, 2, –4), Q (5, 4, –6) and R (9, 8, –10) are collinear. Find the ratio in which Q divides PR.

Q. What is determinate clevage and indeterminate clevage

Q.

The locus of a point which divides the line segment joining the point $(0,-1)$ and a point on the parabola, $x^2=4y$, internally in the ratio $1:2$, is

1. $9x^2-12y=8$

2. $4x^2-3y=2$

3. $x^2-3y=2$

4. $9x^2-3y=2$

Q. The point which divides the line segment joining the points $(6,3)$ and $(-4,5)$ in the ratio $3:2$ externally is

1. $(5,-8)$
2. $(6,9)$
3. $(8,25)$
4. $(-24,9)$

Q. $XY$ – plane divides the line joining the points $A(2,3,-5)$ and $B(-1,-2,-3)$ in the ratio

1. 5:3 Internally
2. 2:1 Internally
3. 5:3 Externally
4. 3:2 Externally

Q. If $x=\frac{a-b}{a+b},y=\frac{b-c}{b+c},z=\frac{c-a}{c+a}$, then the value of $\frac{(1-x)(1-y)(1-z)}{(1+x)(1+y)(1+z)}$ is

Q.

A vector $a$ has magnitude $5$ units and points north east and another vector $b$ has magnitude $5$ units and points north west. Then the magnitude of the vector $\left(a-b\right)$ is?

1. $0$

2. $5\sqrt{2}$

3. $10$

4. $25$

Q. The ratio in which $x-$ axis divides the line segment joining $(3,-4)$ and $(-5,6)$ is

1. $4:3$
2. $2:3$
3. $3:2$
4. $1:3$

Q. Let $N$ be the smallest positive integer such that $N+2N+3N+\dots \dots +9N$ is a number all whose digits are equal. What is the sum of the digits of $N$?

Q. If $(2,1,3),(3,2,5),(1,2,4)$ are the mid points of the sides $BC,CA,AB$ of $△ABC$ respectively, then vertex $A$ is:

1. $(2,3,6)$
2. $(0,2,2)$
3. $(4,1,4)$
4. $(2,5,4)$

Q.

The co-ordinates of the point which divides the join of the points (2, –1, 3) and (4, 3, 1) in the ratio 3 : 4 internally are given by

1. $\frac{2}{7},\frac{20}{7},\frac{10}{7}$

2. $\frac{15}{7},\frac{20}{7},\frac{3}{7}$

3. $\frac{10}{7},\frac{15}{7},\frac{2}{7}$

4. $\frac{20}{7},\frac{5}{7},\frac{15}{7}$