Centroid

Q. Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c),$ $(2,b)$ and $(a,b)$ be $(\frac{10}{3},\frac{7}{3}).$ If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+1=0,$ then the value of ${\alpha }^2+{\beta }^2-\alpha \beta$ is

1. $\frac{71}{256}$
2. $-\frac{71}{256}$
3. $-\frac{69}{256}$
4. $\frac{69}{256}$

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The vertices of a triangle are}(1,a),& \text{(P)}& a+b=13\\ & (2,b)\text{and}(c^2,-3).\text{If the centroid of}& & \\ & \text{the triangle lies on the}x\text{-axis, then}& & \\ \text{(B)}& \text{If}a,b,c(a<b<c)\text{are the roots of}& \text{(Q)}& ab=12\\ & x^3-6x^2+11x-6=0\text{, then}& & \\ \text{(C)}& \text{If}a,b,c\text{lie between 2 and 18 such that}& \text{(R)}& (a,b)=(-5,8)\\ & \text{(i)}a+b+c=25& & \\ & \text{(ii)}2,a,b\text{are in A.P.}& & \\ & \text{(iii)}b,c,18\text{are in G.P., then}& & \\ \text{(D)}& \text{If}a,b,c\text{are in G.P. with common ratio}& \text{(S)}& a+b=8\\ & r(r>1)\text{such that}abc=216\text{and}& & \\ & ab+bc+ca=156\text{, then}& & \\ & & \text{(T)}& a+b=3\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(P)},\text{(Q)}$
2. $\text{(C)}\to \text{(P)},\text{(R)}$
3. $\text{(D)}\to \text{(R)},\text{(S)}$
4. $\text{(D)}\to \text{(Q)},\text{(S)}$

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.

If A(4, -3), B(3, -2) and C(2, 8) are the vertices of a triangle, then its centroid will be

1. (-3, 3)

2. (3, 3)

3. (3, 1)

4. (1, 3)

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The vertices of a triangle are}(1,a),& \text{(P)}& a+b=13\\ & (2,b)\text{and}(c^2,-3).\text{If the centroid of}& & \\ & \text{the triangle lies on the}x\text{-axis, then}& & \\ \text{(B)}& \text{If}a,b,c(a<b<c)\text{are the roots of}& \text{(Q)}& ab=12\\ & x^3-6x^2+11x-6=0\text{, then}& & \\ \text{(C)}& \text{If}a,b,c\text{lie between 2 and 18 such that}& \text{(R)}& (a,b)=(-5,8)\\ & \text{(i)}a+b+c=25& & \\ & \text{(ii)}2,a,b\text{are in A.P.}& & \\ & \text{(iii)}b,c,18\text{are in G.P., then}& & \\ \text{(D)}& \text{If}a,b,c\text{are in G.P. with common ratio}& \text{(S)}& a+b=8\\ & r(r>1)\text{such that}abc=216\text{and}& & \\ & ab+bc+ca=156\text{, then}& & \\ & & \text{(T)}& a+b=3\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(Q)},\text{(R)}$
2. $\text{(A)}\to \text{(R)},\text{(T)}$
3. $\text{(B)}\to \text{(Q)},\text{(T)}$
4. $\text{(B)}\to \text{(R)},\text{(T)}$

Q. For $△ABC$ whose vertices are given by $A(3,6),B(-4,7)$ and $C(10,-7),$ if $A_1,B_1,C_1$ represent the mid points of sides opposite to vertices $A,B,C$ respectively, of $△ABC$ and $A_2,B_2,C_2$ represent the mid points of sides opposite to vertices $A_1,B_1,C_1$ respectively, of $△A_1{B}_1{C}_1$ and so on, then which of the following is/are correct?

1. The centroid of $△A_2{B}_2{C}_2$ is $(3,2)$
2. The centroid of $△A_3{B}_3{C}_3$ is $(3,2)$
3. $\frac{\text{Area (}△A_2{B}_2{C}_2\text{)}}{\text{Area (}△ABC\text{)}}=\frac{1}{16}$
4. $\frac{\text{Area (}△A_3{B}_3{C}_3\text{)}}{\text{Area (}△ABC\text{)}}=\frac{1}{16}$

Q.

If the vertices of a triangle be (a, 1), (b, 3) and (4, c), then the centroid of the triangle will lie on x-axis, if

1. a + c = -4

2. a + b = -4

3. c = -4

4. b + c = -4

Q.

If A(4, -3), B(3, -2) and C(2, 8) are the vertices of a triangle, then its centroid will be

1. (-3, 3)

2. (3, 3)

3. (3, 1)

4. (1, 3)

Q.

If two vertices of a triangle are (6, 4), (2, 6) and its centroid is (4, 6), then the third vertex is

1. (4, 8)

2. (8, 4)

3. (6, 4)

4. (0, 0)

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. Two vertices of a triangle are (3, -5) and (-7, 4). If the centroid of the triangle is (2, -1), then the coordinates of the third vertex are .

1. (10, -2)
2. (-10, -2)
3. (10, 2)
4. (-10, 2)

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 vertex $C$ is

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

Q. A triangle has a vertex at $(1,2)$ and the mid points of the two sides through it are $(–1,1)$ and $(2,3)$. Then the centroid of this triangle is :

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

Q.

If the vertices of a triangle be (a, 1), (b, 3) and (4, c), then the centroid of the triangle will lie on x-axis, if

1. a + c = -4

2. a + b = -4

3. c = -4

4. b + c = -4

Q. Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c),$ $(2,b)$ and $(a,b)$ be $(\frac{10}{3},\frac{7}{3}).$ If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+1=0,$ then the value of ${\alpha }^2+{\beta }^2-\alpha \beta$ is

1. $\frac{71}{256}$
2. $-\frac{69}{256}$
3. $\frac{69}{256}$
4. $-\frac{71}{256}$

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 $\alpha ,\beta ,\gamma$ be the roots of the equation $x^3-12x^2+44x-48=0.$ If the coordinates of the vertices of a triangle are $A(\alpha ,\frac{1}{\alpha }),$ $B(\beta ,\frac{1}{\beta })$ and $C(\gamma ,\frac{1}{\gamma }),$ then the centroid of the $△ABC$ is

1. $(4,\frac{11}{36})$
2. $(\frac{11}{36},4)$
3. $(3,\frac{5}{11})$
4. $(\frac{5}{11},3)$

Q. A point $P$ moves on the line $2x-3y+4=0$. If $Q(1,4)$ and $R(3,-2)$ are fixed points, then the locus of the centroid of $\Delta PQR$ is a line :

1. parallel to x-axis
2. parallel to y-axis
3. with slope $\frac{3}{2}$
4. with slope $\frac{2}{3}$

Q. The locus of centroid of a triangle whose vertices are $(a\cos t,a\sin t),(b\sin t,-b\cos t)$ and $(1,0),$ where $t$ is a parameter, is

1. $(3x-1{)}^2+(3y{)}^2=a^2-b^2$
2. $(3x-1{)}^2+(3y{)}^2=a^2+b^2$
3. $(3x+1{)}^2+(3y{)}^2=a^2+b^2$
4. $(3x+1{)}^2+(3y{)}^2=a^2-b^2$

Q. Let the coordinates of the two vertices of a triangle are $(4,-5),$ $(-6,5)$ respectively. If the coordinates of the centroid are $(3,-1),$ then coordinates of the third vertex are

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

Q. Let $p,q$ and $r$ be the roots of the equation $y^3-3y^2+6y+1=0$. If the vertices of a triangle are $(pq,\frac{1}{pq}),$ $(qr,\frac{1}{qr})$ and $(rp,\frac{1}{rp}),$ then the coordinates of its centroid are

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

Q. Let the coordinates of the two vertices of a triangle are $(4,-5),$ $(-6,5)$ respectively. If the coordinates of the centroid are $(3,-1),$ then coordinates of the third vertex are

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

Q. A variable straight line of slope $4$ intersects the hyperbola $xy=1$ at two points. Find the locus of the point which divides the line segment between these two points in the ratio $1:2$

1. $16x^2+y^2+10xy+2=0$
   
2. $16x^2-y^2+10xy=2$
   
3. $16x^2-y^2-10xy=2$
   
4. $16x^2+y^2+10xy=2$
   

Q.

If the vertex of parabola y = $x^2$-8x + c lies on x - axis, then the value of c is