Section Formula

Q.

The points $A(1,3)$and $C(5,1)$are the opposite vertices of rectangle. The equation of line passing through other two vertices and of gradient $2$, is

1. $2x+y-8=0$

2. $2x-y-4=0$

3. $2x-y+4=0$

4. $2x+y+7=0$

Q.

Let $L_1$ and $L_2$ be the following straight lines. $L_1:\frac{x-1}{1}=\frac{y}{1}=\frac{z-1}{3}$ and $L_2:\frac{x-1}{-3}=\frac{y}{-1}=\frac{z-1}{1}$. Suppose the straight line is $L:\frac{x-\alpha }{l}=\frac{y-1}{m}=\frac{z-\gamma }{-2}$. Lies in the plane containing $L_1$ and $L_2$, and passes through the point of intersection of $L_1$ and

$L_2$. If the line $L$ bisects the acute angle between the lines $L_1$ and $L_2$, then which of the following statements is/are TRUE?

1. $\alpha -\gamma =3$

2. $l+m=2$

3. $\alpha -\gamma =1$

4. $l+m=0$

Q. The ratio in which $y-$ axis divides the line segment joining $(-3,5)$ and $(7,2)$ is

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

Q.

The midpoint of a line segment partitions the line segment into a ratio of:

1. $1:1$

2. $1:2$

3. $2:1$

4. $2:3$

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

Q.

The midpoint of a line segment divides the line segment in the ratio 1:2.

1. True

2. False

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. The three vertices of a parallelogram PQRS are P(3, -1, 2), Q(1, 2, -4), R(-1, 1, 2). Then, the coordinates of the fourth vertex is .

1. (-1, -2, 8)
2. (1, 2, 8)
3. (1, -2, 8)
4. (1, -2, -8)

Q.

If x co-ordinates of a point P of line joining the points Q(2, 2, 1) and R(5, 2, -2) is 4, then the z-coordinates of P is
[RPET 2000]

1. -2

2. -1

3. 1

4. 2

Q. The ratio in which a point $P(2,3)$ divides the line segment joining $A(-2,-7)$ and $B(4,8)$ is

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

Q. If the ratio in which the XY plane divides the line joining the points (2, 4, 5) and (–4, 3, –2) is k: 1, then find the value of 10k

___

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

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

Q. If the vertices of a parallelogram are $A(-2,3),B(3,-1),C(p,q)$ and $D(-1,9)$, then the value of $p+q$ is

Q.

The ends of a rod of length l move on two mutually perpendicular lines. The locus of the point on the rod which divides it in the ratio 1 : 2 is

1. $36x^2+9y^2=4l^2$

2. $36x^2+9y^2=l^2$

3. $9x^2+36y^2=4l^2$

4. 3x+5y-7=0

Q.

xy plane divides the line joining the points $(2,4,5)$and$(-4,3,-2)$ in the ratio

1. 3:5

2. 5:2

3. 1:3

4. 3:4

Q. If $A=(6,-7),B=(-6,5)$ and $P,Q$ are two points on $AB$ such that $AP=PQ=QB$, then which of the following is/are correct?

1. $P=(2,3)$
2. $P=(2,-3)$
3. $Q=(2,-1)$
4. $Q=(-2,1)$

Q. A, B have position vectors $\overrightarrow{a},\overrightarrow{b}$ relative to the origin O and X, Y divide $\overrightarrow{AB}$ internally and externally respectively in the ratio 2 : 1. Then, $\overrightarrow{XY}$ is equal to

1. $\frac{3}{2}(\overrightarrow{b}-\overrightarrow{a})$
2. $\frac{4}{3}(\overrightarrow{a}-\overrightarrow{b})$
3. $\frac{5}{6}(\overrightarrow{b}-\overrightarrow{a})$
4. $\frac{4}{3}(\overrightarrow{b}-\overrightarrow{a})$

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 the join of A(2, 1, 5) and B(3, 4, 3) is divided by the plane 2x+2y-2z-1=0 is .

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

Q.

Which of the follwing statements are correct ?
1. The coordinates of the point R which divides the line

segment joining two points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$

externally in the ratio m:n are $(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n},\frac{m{z}_2+n{z}_1}{m+n})$.

2. If R divides PQ internally in the ratio m:n, then its coordinates

are obtained by replacing n by -n in the statement 1.

1. Only 1

2. Only 2

3. Both 1 and 2

4. None of these

Q.

The point dividing the line joining the points (1, 2, 3) and (3, -5, 6) in the ratio 3 : -5 is

1. $(2,\frac{-25}{2},\frac{3}{2})$

2. $(-2,\frac{25}{2},\frac{-3}{2})$

3. $(2,\frac{25}{2},\frac{3}{2})$

4. None of these

Q.

Which of the following set of points are non- collinear
[MP PET 1990]

1. (1, –1, 1), (–1, 1, 1), (0, 0, 1)

2. (1, 2, 3), (3, 2, 1), (2, 2, 2)

3. (–2, 4, –3), (4, –3, –2), (–3, –2, 4)

4. (2, 0, –1), (3, 2, –2), (5, 6, –4)

Q.

If the points (–1, 3, 2), (–4, 2, –2) and (5, 5, $\lambda$ ) are collinear, then $\lambda$ =

1. – 10

2. 5

3. -5

4. 10

Q.

A(3, 2, 0) , B(5, 3, 2) and C(-9, 6, -3) are three points joining a triangle and AD is bisector of the angle $\angle$ BAC. AD meets BC at the point

1. $(\frac{19}{8},\frac{-57}{16},\frac{17}{16})$

2. $(\frac{-19}{8},\frac{57}{16},\frac{17}{16})$

3. $(\frac{19}{8},\frac{57}{16},\frac{17}{16})$

4. None of these

Q. The ratio in which the plane 2x - 1 = 0 divides the line joining (-2, 4, 7) and (3, -5, 8) is

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

Q. Let the line segment joining $A(6,3)$ and $B(-1,-4)$ is doubled in length by adding equal segments to both the ends. If $P(x_1,y_1)$ and $Q(x_2,y_2)$ are the new end points, then the value of $5(x_1+y_2)+4(x_2+y_1)$ is

Q. Given that the points A(3, 2, -4), B(5, 4, -6) and C(9, 8, -10) are collinear, the ratio in which B divides $\overline{AC}$ is :

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

Q. A, B have position vectors $\overrightarrow{a},\overrightarrow{b}$ relative to the origin O and X, Y divide $\overrightarrow{AB}$ internally and externally respectively in the ratio 2 : 1. Then, $\overrightarrow{XY}$ is equal to

1. $\frac{3}{2}(\overrightarrow{b}-\overrightarrow{a})$
2. $\frac{4}{3}(\overrightarrow{a}-\overrightarrow{b})$
3. $\frac{5}{6}(\overrightarrow{b}-\overrightarrow{a})$
4. $\frac{4}{3}(\overrightarrow{b}-\overrightarrow{a})$

Q. The ratio in which $y-$ axis divides the line segment joining $(-3,5)$ and $(7,2)$ is

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

Q. A rectangle is inscribed in a circle with a diameter lying along the line $3y=x+7$. If the two adjacent vertices of the rectangle are $(-8,5)$ and $(6,5)$, then the area of the rectangle (in sq. units) is :

1. $56$
2. $72$
3. $84$
4. $98$