Section Formula

Q.

Determine the ratio in which the line 3x + y – 9 = 0 divides the segment joining the points (1, 3) and (2, 7). [3 MARKS]

Q. Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}$ equals

1. $\overrightarrow{OA}$
2. $2\overrightarrow{OP}$
3. $3\overrightarrow{OP}$
4. $4\overrightarrow{OP}$

Q. The midpoint of side BC of triangle ABC, with A(1, -4) and the midpoints of the sides through A being (2, -1) and (0, -12) is

Q. Question 15
The line segment joining the points A(3, 2) and B(5, 1) is divided at the point P in the ratio 1:2 and it lies on the line 3x-18y+k=0. Find the value of k.

Q. Find the coordinates of a point P on the line segment joining A(1, 2) and B(6, 7) in such a way that $AP=\frac{2}{5}AB$.

1. $(3,4)$
2. $(-3,-8)$
3. $(-2,5)$
4. $(5,-8)$

Q.

Find the coordinates of the point which divides the line segment joining (-3, 5) and (4, -9) in the ratio 1: 6 internally.

1. $(-2,-3)$

2. $(2,3)$

3. $(-2,3)$

4. $(2,-3)$

Q. If Point P (-4, 6) divides the line segment AB with A(-6, 10) and B(x, y) in the ratio 3:2, find the co-ordinates of B.

1. $(\frac{8}{3},\frac{-10}{3})$
2. $(\frac{-8}{3},\frac{10}{3})$
3. $(\frac{11}{3},\frac{14}{3})$
4. $(\frac{-16}{3},\frac{8}{3})$

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

Q.

Find the point (x, y) that divides the line joining A(3, 6) and B(7, 10) in the ratio 3:1.

1. (8, 9).

2. (4, 5)

3. (6, 9)

4. None of these

Q. Question 9
The point P(5, -3) is one of the two points of trisection of line segment joining the points A(7, -2) and B(1, -5).

Q.

Q.

A(-1, 8), B(4, -2) and C(-5, -3) are the vertices of a triangle, find the equation of the median through (-1, 8).

1. 21x + y + 13 = 0

2. 2x + y - 6 = 0

3. 11x - 4y + 43 = 0

4. x - 9y - 22 = 0

Q. Question 9
Find the coordinates of the points which divide the line segment joining A (- 2, 2) and B (2, 8) into four equal parts.

Q.

Find the point that divides A(2, 4) and B(6, 8) in the ratio a : 1.

1. $(\frac{6+2a}{a+1},\frac{8+4a}{a+1})$

2. $(\frac{6a+1}{a+1},\frac{8a+4}{a+1})$

3. $(\frac{6a+2}{a+1},\frac{8a+4}{a+1})$

4. $(\frac{6a+8}{a+1},\frac{2a+4}{a+1})$

Q.

Midpoint of the line joining the points ($x_1,y_1$) and ($x_2,y_2$) is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$.

1. True

2. False

Q.

The co-ordinates of the point which divides (internally) the line joining the points (– 2, – 2) and (– 5, 7) in the ratio 2 : 1 are

1. (4, – 4)

2. (– 4, 4)

3. (1, – 3)

4. (– 3, 1)

Q. If the end points A and B of the diameter of a circle are such that their abscissas are the roots of the equation x² + 2ax-16=0 and their ordinates are the roots of x2 + 24x-9=0, If a €N and radius R of the circle, with AB as diameter is an integer then, R-a = How to think this

Q. If A(3, 7) and B(7, 9) are two ends of a line segment which is divided by point Q in the ratio 3:4 , then find the co-ordinates of point Q.

1. $(\frac{22}{7},\frac{35}{7})$
2. $(\frac{55}{7},\frac{25}{7})$
3. $(\frac{33}{7},\frac{55}{7})$
4. $(\frac{32}{7},\frac{15}{7})$

Q. Find the distance between the origin and P $(3,\sqrt{5},2)$.

Q.

List - II gives the coordinates of a point P that divides the line segment AB joining the points given in List -I. Match them correctly.
$\begin{array}{cccc}& List-I& & ListII\\ \left(P\right)& A(-1,3)andB(-5,6)Internallyintheratio1:2& \left(1\right)& (7,3)\\ \left(Q\right)& A(-2,1)andB(1,4)internallyintheratio2:1& \left(2\right)& (0,3)\\ \left(R\right)& A(1,7)andB(3,4)internallyintheratio3:2& \left(3\right)& (\frac{11}{5},\frac{26}{5})\\ \left(S\right)& a(4,-3)andB(8,5)internallyintheratio3:1& \left(4\right)& (-\frac{7}{3},4)\end{array}$

1. P- 4, Q-2, R-3, S-1

2. P- 3, Q-2, R-4, S-1

3. P- 1, Q-4, R-3, S-2

4. P- 3, Q-1, R-2, S-4

Q.

Find the point which divides the line segment joining the points (2, -6) and (3, 6) in the ratio 2 : 3.

1. $(\frac{12}{5},-\frac{6}{5})$

2. $(\frac{12}{5},\frac{6}{5})$

3. $(-\frac{12}{5},-\frac{6}{5})$

4. $(-\frac{12}{5},\frac{6}{5})$

Q.

$L1:x-2y+10=0$

$L2:x+2y-6=0$

Ratio in which the point of intersection of line L1 and L2 divides the line segment AB of L1 is ?

1. $2:3$

2. $5:3$

3. $1:1$

4. $2:5$

Q. The ratio in which $\hat{i}+2\hat{j}+3\hat{k}$ divides the join of $-2\hat{i}+3\hat{j}+5\hat{k}$ and $7\hat{i}-\hat{k}$ is

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

Q. Find the coordinates of the point P which divides segment JL externally in the ratio m:n in the following examples.
(i) J(4, -5), L(-6, 7), m:n = 3:5

Q. Which side will be present between the vertices A and B of a triangle ABC?

1. BC
2. AC
3. AB
4. CA

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

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Q. If the vectors $PQ=-3i+4j+4k$ and $PR=5i-2j+4k$ are the sides of a $△PQR$, then the length of the median through $P$ is

1. $\sqrt{14}$
2. $\sqrt{15}$
3. $\sqrt{17}$
4. $\sqrt{18}$
5. $\sqrt{19}$

Q. The coordinates of centroid of triangle formed by lines 4x+7y =28 whenx=0, y=0

Q.

Find the area of the triangle formed by joining the mid points of the sides of the triangle formed with coordinates A (-4, 3), B (2, 3) and C (4, 5).

1. 0.5

2. 1

3. 1.5

4. 2

Q. Find the point which divides internally and externally, the line joining $(-1,2)$ to $(4,-5)$ in the ratio $2:3$.