Centroid

Q. In a triangle $ABC$, coordinates of $A$ are $(1,2)$ and the equations of the medians through $B$ and $C$ are respectively, $x+y=5$ and $x=4$. Then area of $△ABC$ (in sq. units) is :

1. $12$
2. $4$
3. $5$
4. $9$

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 $\left(\frac{10}{3},\frac{7}{3}\right)$. 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. 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. $(1,\frac{7}{3})$
2. $(\frac{1}{3},\frac{5}{3})$
3. $(\frac{1}{3},1)$
4. $(\frac{1}{3},2)$

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

Q.

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

1. $3x-2y=3$

2. $2x+3y=9$

3. $2x-3y=7$

4. $3x+2y=5$

Q.

The centroid of an equilateral triangle is (0, 0). If two vertices of the triangle lie on x + y = 2$\sqrt{2}$ , then one of them will have its coordinates as

1. $(\sqrt{2}+\sqrt{6},\sqrt{2}-\sqrt{6})$
2. $(\sqrt{2}+\sqrt{3},\sqrt{2}-\sqrt{3})$
3. $(\sqrt{2}+\sqrt{5},\sqrt{2}-\sqrt{5})$
4. $(\sqrt{2}+\sqrt{6},\sqrt{2}+\sqrt{6})$

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

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 centroid of a triangle is $(2,7)$ and two of its vertices are $(4,8)$ and $(-2,6).$ The third vertex is

1. (7, 4)

2. (7, 7)

3. (0, 0)

4. (4, 7)

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

Q.

If the points (1, -1), (2, -1) and (4, -3) are the mid-points of the sides of a triangle, then write the coordinates of its centroid.

Q.

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2).

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. $(9,7)$
4. $(7,6)$

Q. A quadratic function $y=p{x}^2+qx+r$, where $p,q$ and $r$ are distinct real numbers satisfies the below conditions :
$\left(1\right)$ The graph of $y$ passes through the point $(1,6)$.
$\left(2\right)\frac{1}{p},\frac{1}{q},\frac{1}{r}$ form an arithmetic progression.
$\left(3\right)p,r,q$ are in geometric progression.

Which of the following options is/are correct?

1. Sum of the roots of $y=0$ is $-4$.
2. Product of the roots of $y=0$ is $-2.$
3. There exist $2$ real and distinct roots of $y=0$
4. There exist $2$ real and equal roots of $y=0$.

Q. From any point P lying in first quadrant on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ PN is drawn perpendicular to the major axis such that N lies on major axis. Now PN is produced to the point Q such that NQ equals to PS, where S is a focus. The point Q lies on which of the following lines

1. 2y-3x-25=0
2. 3x+5y+25=0
3. 2x-5y+25=0
4. 2x-5y-25=0

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

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. For the triangle formed by the point $(4,8),(2,8),(4,12)$

1. Orthocentre $\equiv (5,7)$
2. Orthocentre $\equiv (4,8)$
3. Circumcentre $\equiv (3,10)$
4. Centroid $\equiv (\frac{10}{3},\frac{28}{3})$

Q.

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

1. $2\sqrt{2}$

2. $\sqrt{2}$

3. 2

4. 1

Q.

How do you find the terminal point $p(x,y)$ on the unit circle determined by the given value of $t=-\frac{3\mathrm{\pi }}{4}$ ?

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

Q. Consider a $△ABC,$ whose vertices are $A(-2,3,-4),B(4,2,1)$ and $C(1,1,0),$ then the distance (in units) of its centroid from origin is

1. $2$
2. $\sqrt{6}$
3. $4$
4. $2\sqrt{5}$

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. 43. Consider 3 non collinear points A, B, C with coordinates (0, 6), (5, 5) and (-1, 1) respectively.Equation of a line tangent to the circle circumscribing the triangle ABC and passing through the origin is where x < y is 1. 11 2. 7 3. 9 4. 13

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.

Find the coordinates of the incentre and centroid of the triangle whose sides have the equations $3x-4y=0,12y+5x=0$ and $y-15=0.$

Q. The area of an equilateral triangle whose two vertices are (1, 0) and (3, 0) and third vertex lying in the first quadrant is-

1. $\sqrt{3}/4$
2. $\sqrt{3}/2$
3. $\sqrt{3}$
4. None of these

Q. One vertex of the equilateral triangle with centriod at origin and one side as $x+y-2=0$ is

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