Centroid

Q.

In what ratio, the line joining (-1, 1) and (5, 7) is divided by the line x+y=4?

Q. A straight rod of length 3l units slides with its ends A, B always on the x and y axes respectively then the locus of centroid of $△OAB$ is

1. 
2. 
3. 
4. 

Q.

If the vertices of a triangle be (a, b - c), (b, c - a) and (c, a - b), then the centroid of the triangle lies

1. At origin

2. On x-axis

3. On y-axis

4. None of these

Q. if a(9, -9) , (1, -3) are the end points of hypoteneuse of a right angled isosceles triangle , then the third vertecs is

Q.

Let A(2, - 3) and B(-2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1

then the locus of the vertex C is the line

1. 

2. 

3. 

4. 

Q. 25. S and T are the focii of an ellipse and B is the end of minor axis .If STB is an equilateral triangle then the eccentricity of the ellipse is

Q.

The ratio in which line segment joining the points $(-3,10)$ and $(6,-8)$ is divided by $(-1,6)$ is

1. $\frac{2}{7}$

2. $\frac{7}{2}$

3. 2

4. $\frac{1}{2}$

Q. Prove that the distance between the points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$ is given by
$PQ=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$

Q. Let $O(0,0),P(3,4),Q(6,0)$ be the vertices of the triangle OPQ. The point R inside the tringle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

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

Q. If the centroid and a vertex of an equilateral triangle are $(2,3)$ and $(4,3)$ respectively, then the other two vertices of the triangle are

1. $(1,3+\sqrt{3})\text{and}(1,3-\sqrt{3})$
2. $(2,3+\sqrt{3})\text{and}(2,3-\sqrt{3})$
3. $(-1,3+\sqrt{3})\text{and}(-1,3-\sqrt{3})$
4. $(1,3+2\sqrt{3})\text{and}(1,3-2\sqrt{3})$

Q. Let $O(0,0),P(3,4),Q(6,0)$ be the vertices of the triangle OPQ. The point R inside the tringle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

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

Q. The vertices of a triangle are (6, 0), (0, 6) and (6, 6). The distance between its circumcentre and centroid is

(a) $2\sqrt{2}$
(b) 2
(c) $\sqrt{2}$
(d) 1

Q. Let A (2, -3) and B(-2, 1) be the vertices of a tirangle ABC. If the centroid of this triangle moves on the line $2x+3y=1,$ then the locus of the vertex C is the line

1. 2x+3y=9
2. 5x-3y=7
3. 3x+2y=5
4. 3x-2y=3

Q. If $C$ be the centroid of the triangle having vertices $(3,-1),(1,3)$ and $(2,4)$. Let $P$ be the point of intersection of the lines $x+3y-1=0$ and $3x-y+1=0$, then the line passing through the points $C$ and $P$ also passes through the point :

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

Q.

Let A(2, - 3) and B(-2, 1) be vertices of a triangle ABC. If the centroid of this triangle moves on the line 2x + 3y = 1 then the locus of the vertex C is the line

1. 

2. 2x + 3y = 9

3. 3x - 2y = 3

4. 3x + 2y = 5

Q. If $(\frac{5}{3},3)$ is the centroid of a triangle and its two vertices are $(0,1)$ and $(2,3)$, then the coordinates of the third vertex are

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

Q. A straight rod of length 3l units slides with its ends A, B always on the x and y axes respectively then the locus of centroid of $△OAB$ is

1. $x^2+y^2=3l^2$
2. $x^2+y^2=l^2$
3. $x^2+y^2=4l^2$
4. $x^2+y^2=2l^2$

Q. Q.12. If a circle touches the straight lines $4x-3y+10=0,4x-3y-30=0$ and one of its normal is 2x + y = 0, then the centre of the circle is at
(a) (- 3, 6)
(b) (3, - 6)
(c) (- 1, 2)
(d) (1, - 2)

Q. The co-ordinates of the vertices of a triangle are (4, -3), (-5, 2) and ( x, y). If the centroid of the triangle is at the origin, Show that x= y = 1?