Relation between O,G,C

Q. If the orthocenter and centroid of a triangle are (–3, 5) and (3, 3) respectively then the circumcenter is

1. (6, 2)
2. (0, 4)
3. (6, -2)
4. (0, 8)

Q. If $A(\cos \alpha ,\sin \alpha ),$ $B(\cos \beta ,\sin \beta ),$ $C(\cos \gamma ,\sin \gamma )$ are the vertices of $△ABC$ and $H,G,S$ are the orthocentre, centroid and circumcentre of $△ABC$ respectively, then which of the following is/are true?

1. $S\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
2. $G\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
3. $H\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
4. $S\equiv (0,0)$

Q. The vertices of an acute angled triangle are $A(x_1,x_1\tan \alpha ),B(x_2,x_2\tan \beta )$ and $C(x_3,x_2\tan \gamma ).$ If origin is the circumcentre of $△ABC$ and $H(a,b)$ be its orthocentre, then $\frac{b}{a}$ equals to
(where $x_1,x_2,x_3$ are positive)

1. $\frac{\cos \alpha +\cos \beta +\cos \gamma }{\cos \alpha \cos \beta \cos \gamma }$
2. $\frac{\sin \alpha +\sin \beta +\sin \gamma }{\sin \alpha \sin \beta \sin \gamma }$
3. $\frac{\sin \alpha +\sin \beta +\sin \gamma }{\cos \alpha +\cos \beta +\cos \gamma }$
4. $\frac{\tan \alpha +\tan \beta +\tan \gamma }{\tan \alpha \tan \beta \tan \gamma }$

Q.

The position vectors of the vertices of an equilateral triangle, whose orthocentre is at the origin, then?

1. $a+b+c=0$

2. $a^2=b^2+c^2$

3. $a+b=c$

4. None of these

Q. If a triangle has its orthocentre at $(1,1)$ and circumcentre at $(\frac{3}{2},\frac{3}{4}),$ then the coordinates of the centroid of the triangle are

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

Q. Find the position vector of a point R which divides the line joining the two points P and Q with position vectors $\overrightarrow{\mathrm{OP}}=2\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{\mathrm{OQ}}=\overrightarrow{a}-2\overrightarrow{b}$, respectively in the ratio 1 : 2 internally and externally. [NCERT EXEMPLAR]

Q. If $A(7,9),B(3,-7)$ and $C(-3,3)$ are the vertices of a triangle and $S(\alpha ,\beta ),O(l,m)$ are its circumcentre and orthocentre respectively, then the square of the distance between $P(m,\alpha )$ and $Q(\beta ,l)$ is

Q. If the orthocenter of a triangle is $(6,3)$ and centroid is $(2,5)$, then find the circumcenter of the triangle.

1. $(1,2)$
2. $(1,3)$
3. $(0,5)$
4. $(0,6)$

Q. If $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ are the vertices of an equilateral triangle and $(x_1-4{)}^2+(y_1-5{)}^2=(x_2-4{)}^2+(y_2-5{)}^2=(x_3-4{)}^2+(y_3-5{)}^2,$ then the value of $(y_1+y_2+y_3)-(x_1+x_2+x_3)$ is

Q. Points $O,A,B,C,\dots$ are shown in figure where $OA=2AB=4BC=\dots$ so on. Let $A$ be the centroid of a triangle whose orthocentre and circumcentre are $(2,4)$ and $(\frac{7}{2},\frac{5}{2})$ respectively. If an insect starts moving from the point $O(0,0)$ along the straight line in zig-zag fashion and terminates ultimately at point $P(\alpha ,\beta ),$ then the value of $\alpha +\beta$ is

Q.

Find the direction cosines of the vector joining the points $A(1,2,-3)andB(-1,-2,1)$, directed from A to B.

Q. If the orthocentre, centroid and the circumcentre of a triangle $ABC$ coincide with each other and if the length of side $AB$ is $8\sqrt{3}$, then the length of the altitude through the vertex $A$ is

Q. If $l,m,n$ denote the sides of pedal triangle opposite to the vertices $A,B,C$ respectively of triangle$ABC.$ Then the value of $\frac{l}{a^2}+\frac{m}{b^2}+\frac{n}{c^2}$ is :

1. $\frac{a^2+b^2+c^2}{a^3+b^3+c^3}$
2. $\frac{a^2+b^2+c^2}{2abc}$
3. $\frac{a^3+b^3+c^3}{abc(a+b+c)}$
4. $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Q. Consider an isosceles triangle $ABC$ with $AB=4,$ $BC=5,$ $AC=4$, having $O,G,S$ as orthocentre, centroid and circumcentre respectively, then the area (in sq. units) of $△OGS$ is

Q.

The co-ordinates of the vertices of the triangle are A(-2, 3, 6), B(-4, 4, 9) and C(0, 5, 8). The direction cosines of the median BE are:

1. 

2. 

3. 

4. 

Q.

If a triangle has its orthocenter at (1, 1) and circumcenter at $(\frac{3}{2},\frac{3}{4})$ , then the coordinates of the centroid of the triangle are

1. $(\frac{4}{3},-\frac{5}{6})$

2. $(\frac{4}{3},\frac{5}{6})$

3. $(-\frac{4}{3},\frac{5}{6})$

4. $(-\frac{4}{3},-\frac{5}{6})$

Q. If $A(\cos \alpha ,\sin \alpha ),$ $B(\cos \beta ,\sin \beta ),$ $C(\cos \gamma ,\sin \gamma )$ are the vertices of $△ABC$ and $H,G,S$ are the orthocentre, centroid and circumcentre of $△ABC$ respectively, then which of the following is/are true?

1. $S\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
2. $G\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
3. $H\equiv \left(\sum \cos \alpha ,\sum \sin \alpha \right)$
4. $S\equiv (0,0)$

Q. The coordinates of the point $P$ on the line $2x+3y+1=0$ such that $|PA-PB|$ is maximum, where $A(2,0)$ and $B(0,2)$ is

1. $(4,-3)$
2. $(7,-5)$
3. $(10,-7)$
4. $(-8,5)$

Q. The vertices of an acute angled triangle are $A(x_1,x_1\tan \alpha ),B(x_2,x_2\tan \beta )$ and $C(x_3,x_2\tan \gamma ).$ If origin is the circumcentre of $△ABC$ and $H(a,b)$ be its orthocentre, then $\frac{b}{a}$ equals to
(where $x_1,x_2,x_3$ are positive)

1. $\frac{\cos \alpha +\cos \beta +\cos \gamma }{\cos \alpha \cos \beta \cos \gamma }$
2. $\frac{\sin \alpha +\sin \beta +\sin \gamma }{\sin \alpha \sin \beta \sin \gamma }$
3. $\frac{\tan \alpha +\tan \beta +\tan \gamma }{\tan \alpha \tan \beta \tan \gamma }$
4. $\frac{\sin \alpha +\sin \beta +\sin \gamma }{\cos \alpha +\cos \beta +\cos \gamma }$

Q. If ($\alpha ,\beta$), $(\overline{x},\overline{y})and(u,v)$are respectively coordinates of the circumcentre, centroid and orthocentre of a triangle.

1. 3$\overline{x}=2\alpha +uand3\overline{y}=2\beta +v$
2. 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta -v$
3. 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta +v$
4. 3$\overline{x}=2\alpha +uand3\overline{y}=2\beta -v$

Q. If ($\alpha ,\beta$), $(\overline{x},\overline{y})and(u,v)$are respectively coordinates of the circumcentre, centroid and orthocentre of a triangle.

1. 3$\overline{x}=2\alpha +uand3\overline{y}=2\beta +v$
2. 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta -v$
3. 3$\overline{x}=2\alpha -uand3\overline{y}=2\beta +v$
4. 3$\overline{x}=2\alpha +uand3\overline{y}=2\beta -v$

Q.

In the following example find the distance of each of the given points from the corresponding given plane:

$\begin{array}{cc}point& Plane\\ (2,3,-5)& x+2y-2z=9\end{array}$

Q. In $\Delta ABC,$ the ratio $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ is always equal to:
(where for $△ABC$ usual notations are used and $h_1,h_2,h_3$ are altitudes from respective vertices.)

1. $\frac{(abc{)}^{\frac{1}{2}}}{(h_1{h}_2{h}_3{)}^{\frac{1}{6}}}$
2. $\frac{(abc{)}^{\frac{4}{3}}}{\left(h_1{h}_2{h}_3\right)}$
3. $\frac{(abc{)}^{\frac{1}{3}}}{(h_1{h}_2{h}_3{)}^{\frac{2}{3}}}$
4. $\frac{(abc{)}^{\frac{2}{3}}}{(h_1{h}_2{h}_3{)}^{\frac{1}{3}}}$

Q. If the orthocenter and circumcentre of a triangle are (0, 0) and (3, 6) respectively then the centroid of the triangle is

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

Q. Find the equation of the chord of contact of tangents from the point $(1,1)$ for the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1.$

Q. If $G$ is the centroid of the $△ABC$ and $G^{\prime }$ is the centroid of the $△A^{\prime }{B}^{\prime }{C}^{\prime }$, then $A{A}^{\prime }+B{B}^{\prime }+C{C}^{\prime }$ is

1. $3G{G}^{\prime }$
2. None of these
3. $G{G}^{\prime }$
4. $2G{G}^{\prime }$

Q.

Find the lengths of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and (6, 0, 0).

Q. Let $\alpha =\cos \frac{2\pi }{7}+\cos \frac{4\pi }{7}+\cos \frac{6\pi }{7}$ and $\beta ={\sin }^2\frac{\pi }{8}+{\sin }^2\frac{3\pi }{8}+{\sin }^2\frac{5\pi }{8}+{\sin }^2\frac{7\pi }{8}$. The angle between the two lines whose slopes are $\alpha ,\beta$ is

1. $\frac{\pi }{4}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{2}$

Q. Tangents are drawn to hyperbola $\frac{x^2}{25}-\frac{y^2}{9}=1$ at points whose eccentric angles are $30°$ and $60°$. Find point of intersection of tangents at these points.

Q.

If a triangle has its orthocenter at (1, 1) and circumcenter at $(\frac{3}{2},\frac{3}{4})$ , then the coordinates of the centroid of the triangle are

1. $(\frac{4}{3},-\frac{5}{6})$

2. $(\frac{4}{3},\frac{5}{6})$

3. $(-\frac{4}{3},\frac{5}{6})$

4. $(-\frac{4}{3},-\frac{5}{6})$