Internal Division

Q. The point A divides the join of P(-5, 1) and Q(3, 5) in the ratio k:1. Find the two values of k for which the area of $\Delta ABC$ where B is (1, 5) and C is (7, -2) is equal to 2 units.

1. undefined
2. undefined
3. undefined
4. undefined

Q.

A rod of length l slides between the two perpendicular lines.Find the locus of the point on the rod which divedes it in the ratio 1 : 2.

Q.

A variable line passes through a fixed point $P$ The algebraic sum of the perpendicular drawn from $\left(2,0\right),\left(0,2\right)$ and $\left(1,1\right)$ on the line is zero, then the coordinates of the $P$ are

1. $\left(1,-1\right)$

2. $\left(1,1\right)$

3. $\left(2,1\right)$

4. $\left(2,2\right)$

Q.

Find the point of the intersection of the medians of a triangle whose vertices are $(-1,0)$, $(5,-2)$ and $(8,2)$

Q.

Equation of the line which passes through the point $(-4,3)$ and the portion of the line intercepted between the axes is divided internally in the ratio $5:3$ by this point, is

1. $20x+9y+96=0$
2. $9x-20y+96=0$
3. None of these
4. $9x+20y+96=0$

Q. A rod of length $l$ moves such that its ends $A$ and $B$ always lie on the lines $3x-y+5=0$ and $y+5=0$ respectively. The locus of the point $P,$ which divides $AB$ internally in the ratio $2:1,$ is $l^2=\frac{1}{k}(ax-by-5{)}^2+9(y+5{)}^2.$ Then

1. $k=4,a+b=6$
2. $k=3,a+b=5$
3. $k=4,a+b=0$
4. $k=3,a+b=4$

Q. The coordinates of the point(s), which trisect(s) the line segment joining the points $(1,-2)$ and $(-3,4),$ are

1. $(0,\frac{-1}{2})$
2. $(\frac{-1}{3},0)$
3. $(-2,\frac{5}{2})$
4. $(\frac{-5}{3},2)$

Q. If P and Q are two points on the line 4x + 3y + 30 = 0 such that OP = OQ = 10, where O is the origin, then the area of the $\Delta$ OPQ is

1. 48
2. 16
3. 32
4. None of these

Q. $P$ and $Q$ are two points lying on the line joining the points $A(12,8)$ and $B(-2,6)$ such that $AP=PQ=QB.$ Then the midpoint of $PQ$ is

1. $(7,1)$
2. $(9,3)$
3. $(6,4)$
4. $(5,7)$

Q. If coordinates of the centre and one end of a diameter of a circle are $(7,3)$ and $(5,-7)$ respectively, then the coordinates of the other end of the diameter are

1. $(6,-2)$
2. $(9,13)$
3. $(-2,6)$
4. $(12,-4)$

Q. The equation of the plane mid-parallel to the planes $2x-3y+6z-7=0$ and $2x-3y+6z+7=0$ is

1. $2x-3y+6z=6$
2. $2x-3y+6z=0$
3. $2x-3y+6z=14$
4. $2x-3y+6z=8$

Q. The vertices of $\Delta PQR$ are $P(2,1),Q(-2,3)$ and $R(4,5)$. Find equation of the median through the vertex $R$.

Q.

Find the coordinates of the points which tisect the line

segment joining the points P(4, 2, -6) and Q (10, -16, 6)

Q. A straight line through the origin O meets the parallel lines 4x+2y = 9 and 2x+y+ 6 = 0 at points P and Q respectively. Then the point O divides the segment PQ in ratio

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

Q.

If A (-1, 8), B(4, -2) and C(-5, -3) are the vertices of a triangle. Median through(-1, 8) intersect line segment BC at D. Find the co-ordinates of point D.

1. 

2. 

3. 

4. 

Q. A straight line through the origin O meets the parallel lines 4x+2y = 9 and 2x+y+ 6 = 0 at points P and Q respectively. Then the point O divides the segment PQ in ratio

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

Q. $1.4+2.5+...+n(n+3)=$

1. $\frac{n(n+3)(n+5)}{9}$
2. $\frac{n(n+1)(n+5)}{3}$
3. $\frac{n(n+3)(n+9)}{12}$
4. $\frac{n(n+5)(n+7)}{6}$

Q. If a point $R(4,y,z)$ lies on the line segment joining the points $P(2,-3,4)$ and $Q(8,0,10)$, then the distance of $R$ from the origin is:

1. $6$
2. $2\sqrt{14}$
3. $\sqrt{53}$
4. $2\sqrt{21}$