External Division

Q. The harmonic conjugate of the point $R(2,4)$ with respect to the points $P(2,2)$ and $Q(2,5)$ is

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

Q. The harmonic conjugate of $(2,3,4)$ w.r.t the points $(3,-2,2)$ and $(6,-17,-4)$ is:

1. $(\frac{-1}{2},\frac{1}{3},\frac{-1}{4})$
2. $(\frac{1}{2},\frac{1}{3},\frac{1}{4})$
3. $(\frac{18}{7},2,\frac{1}{5})$
4. $(\frac{18}{5},-5,\frac{4}{5})$

Q. The coordinates of the point(s) which divide(s) the line segment joining the point $(5,-2)$ and $(9,6)$ in the ratio $3:1,$ are

1. $(11,10)$
2. $(8,4)$
3. $(10,11)$
4. $(4,8)$

Q. The ratio in which the point $(2,y)$ divides the line segment joining $(4,3)$ and $(6,3)$ is

1. $1:2$ externally
2. $1:2$ internally
3. $3:2$ externally
4. $3:2$ internally

Q. If the line segment joining $(2,-3)$ and $(5,6)$ is divided by the $y-$axis, then

1. the point of division is $(0,-9)$
2. the point of division is $(0,9)$
3. the ratio is $2:5$ externally
4. the ratio is $2:5$ internally

Q. The harmonic conjugate of the point $C(-6,-2)$ with respect to the points $A(-10,0)$ and $B(0,-5)$ is

1. $(10,-20)$
2. $(20,10)$
3. $(-30,10)$
4. $(10,30)$

Q.

The point which divides the join of $(1,2)$ and $(3,4)$ externally in the ratio 1 : 1

1. cannot be found

2. lies in the III quadrant

3. lies in the II quadrant

4. lies in the I quadrant

Q. The coordinates of the point which divides the line segment joining (4, –3) and (9, 7) in the ratio 3 : 2 externally is .

1. (19, 27)
2. (7, 3)
3. (35, 15)
4. $(\frac{19}{5},\frac{27}{5})$

Q.

Find the slope of a line, which passes through the origin, and the mid-point of

the line segment joining the points P (0, –4) and B (8, 0).

Q. If $p,q,r$ are statements, with truth values T, F, T respectively, then the truth value of $(\sim p\vee q)\wedge \sim r\Rightarrow p$ is T

1. True
2. False

Q. If P (2, 6) is the mid-point of the line segment joining A (6, 5) and B (4, y), find y.

Q. The harmonic conjugate of the point $C(-6,-2)$ with respect to the points $A(-10,0)$ and $B(0,-5)$ is

1. $(20,10)$
2. $(10,30)$
3. $(10,-20)$
4. $(-30,10)$

Q. If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular, find the value of $k$

Q. Find the ratio in which the point $P(5,4,-6)$ divides the line segment joining the points $A(3,2,-4)$ and $B(9,8,-10)$. Also, find the harmonic conjugate of $P$.

Q. The coordinates of the point which divides the line segment joining (4, –3) and (9, 7) in the ratio 3 : 2 externally is .

1. (19, 27)
2. (7, 3)
3. (35, 15)
4. $(\frac{19}{5},\frac{27}{5})$

Q. Find the equation of line passing through the points A (0, 6, −9) and B (−3, −6, 3). If D is the foot of perpendicular drawn from a point C (7, 4, −1) on the line AB, then find the coordinates of the point D and the equation of line CD.

Q. A (1, 0, 4), B (0, −11, 3), C (2, −3, 1) are three points and D is the foot of perpendicular from A on BC. Find the coordinates of D.

Q. Let $G$ be the centroid of $△ABC$. If $\overrightarrow{AB}=\overrightarrow{a}$ and $\overrightarrow{AC}=\overrightarrow{b},$ then $\overrightarrow{AG}$, in terms of $\overrightarrow{a}$ and $\overrightarrow{b}$ is

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

Q. The harmonic conjugate of the point $C(-6,-2)$ with respect to the points $A(-10,0)$ and $B(0,-5)$ is

1. $(20,10)$
2. $(10,30)$
3. $(10,-20)$
4. $(-30,10)$

Q. Find the foot of perpendicular from the point (2, 3, 4) to the line $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}.$ Also, find the perpendicular distance from the given point to the line.

Q. Pont P divides the line segment joining the points A ( 2, 1 ) and B ( 5, -8 ) such that AP / AB = 1 / 3. If P lies on the line 2x - y + k = 0, find the value of k.

Q. If the points $A(a,b)$, $B(-a,-b)$ and $P(a^2,ab)$ are collinear then the ratio in which $P$ divides $\overline{AB}$ is

1. $1+a:1-a$
2. $a:1$
3. $1-a:1+a$
4. $1:a$

Q. If the direction ratios of two parallel lines are $4,-3,-1$ and $p+q,1+q,2$ then, find the values of $p$ and $q$.

Q. The ratio in which the point $(2,y)$ divides the line segment joining $(4,3)$ and $(6,3)$ is

1. $1:2$ externally
2. $1:2$ internally
3. $3:2$ externally
4. $3:2$ internally

Q. Find the equation of line passing through the points A (0, 6, −9) and B (−3, −6, 3). If D is the foot of perpendicular drawn from a point C (7, 4, −1) on the line AB, then find the coordinates of the point D and the equation of line CD.