Circumcentre

Q. Let $ABC$ be the triangle with $AB=1,AC=3$ and $\angle BAC=\frac{\pi }{2}$. If a circle of radius $r>0$ touches the sides $AB,AC$ and also touches internally the circumcircle of the triangle $ABC,$ then the value of $r$ is

Q.

If the vertices of a triangle be (0, 0), (6, 0) and (6, 8), then its incentre will be

1. (1, 2)

2. (2, 1)

3. (4, 2)

4. (2, 4)

Q. The circumcentre of the triangle formed by $(2,-5),$ $(2,7),$ $(4,7)$ is

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

Q. If $S$ is the circumcentre of $△ABC$, where $A=(1,3),B=(-3,5),C=(5,-1)$, then

1. coordinates of $S$ will be $(-8,-10)$
2. coordinates of $S$ will be $(-4,-5)$
3. radius of circumcircle will be $5\sqrt{10}$ units
4. radius of circumcircle will be $5\sqrt{5}$ units

Q. Find the slope of the lines :
$\left(i\right)$ Passing through the points $(3,-2)\text{and}(-1,4)$
$\left(ii\right)$ Passing through the points $(3,-2)\text{and}(7,-2)$
$\left(iii\right)$ Passing through the points $(3,-2)\text{and}(3,4)$
$\left(iv\right)$ Making inclination of ${60}^{\circ }$ with the positive direction of $x-$axis.

Q. A line passes through $(x_1,y_1)\text{and}(h,k).$ If slope of the line is $m$, show that $k–y_1=m(h–x_1)$.

Q.

The verties of a triangle are O (0, 0), A(a, 0)andB(0, b). Write the coordinates of its circumcentre.

Q.

Write the coordinates of the circumcentre of a triangle whose centroid and orthoentre are at (3, 3) and (-3, 5) respectively.

Q. Find the incentre of the triangle whose vertices are A(1, 2, 3), B(1, 5, 3) and C(5, 2, 3).

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

Q. Find the equation of the line which is passing through $(2,2\sqrt{3})$ and inclined with the $x$-axis at an angle of ${75}^{\circ }$.

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 $(x_1-2{)}^2+(y_1-3{)}^2=(x_2-2{)}^2+(y_2-3{)}^2=(x_3-2{)}^2+(y_3-3{)}^2$, then

1. Coordinates of orthocentre is $(0,0)$
2. Coordinates of circumcentre is $(2,3)$
3. value of $(x_1+x_2+x_3)+3(y_1+y_2+y_3)$ is $33$
4. value of $2(x_1+x_2+x_3)-(y_1+y_2+y_3)$ is $3$

Q. If the vertices of a triangle are $P(8,6),Q(8,-2)$ and $R(2,-2)$, then which of the following is/are correct?

1. Coordinates of circumcentre of triangle PQR is $(5,2)$
2. Coordinates of circumcentre of triangle PQR is $(2,5)$
3. Circumradius of triangle PQR is $5$ units
4. Circumradius of triangle PQR is $\sqrt{37}$ units

Q. Line through the points $(–2,6)\text{and}(4,8)$ is perpendicular to the line through the points $(8,12)\text{and}(x,24)$. Find the value of $x$.

Q. If $A=(6,0),B=(0,8),$ and the circumcenter of the $△OAB$ where $O$ is the origin, is $(\alpha ,\beta ),$ then the value of $\alpha +\beta$ is

Q. The slope of the line passing through points $(3,2)$ and $(2,5)$ is

1. $\frac{1}{3}$
2. $-3$
3. $-\frac{1}{3}$
4. $3$

Q. The slope of the line passing through points $(3,2)$ and $(2,5)$ is

1. $\frac{1}{3}$
2. $-\frac{1}{3}$
3. $3$
4. $-3$

Q. The area of the region included between the parabola $y=\frac{3x^2}{4}$ and the line 3x – 2y + 12 = 0 is

1. $\frac{4}{3}$ sq.unit
2. $\frac{2}{3}$ sq.unit
3. $\frac{21}{2}$ sq.unit
4. 27 sq. unit

Q. If a line makes an angle of ${30}^{\circ }$ with the positive direction of $y-$ axis measured in anticlockwise direction, then the slope of the line is

1. $\sqrt{3}$
2. $-\sqrt{3}$
3. $\frac{1}{\sqrt{3}}$
4. $-\frac{1}{\sqrt{3}}$

Q. Find the angle between the $x-$axis and the line joining the points $(3,–1)$ and $(4,–2)$.

Q. $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ are the vertices of a triangle. If P (x, y) is the circumcentre of the triangle, then
PA=PB=PC

1. True

2. False

Q. Consider a right angled triangle $ABC$


If the sides of the triangle are $a=6,b=10,c=8$ units, then the distance between incentre and circumcentre will be

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

Q.

$InQ>No.1,\text{Write the distance between the circumcentre and orthocentre of}\Delta OAB.$

Q. The circumcentre of the triangle formed by $(2,-5),$ $(2,7),$ $(4,7)$ is

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

Q.

If $A(x_1,y_1)$, $B(x_2,y_2)$ and $C(x_3,y_3)$ are the vertices of a triangle, then the excentre opposite to B is

1. None of these

2. 

3. 

4. 

Q. The equation of the line which makes an angle of ${15}^{\circ }$ with positive $x$-axis and cuts-off an intercept of $3$ unit on the negative $y$-axis is

1. $y-(2-\sqrt{3})x+3=0$
2. $y+(2-\sqrt{3})x+3=0$
3. $y-(2-\sqrt{3})-3=0$
4. $y-(2+\sqrt{3})x+3=0$

Q. A straight line $L$ through the point $(3,-2)$ is inclined at an angle of ${60}^{\circ }$ to the line $\sqrt{3}x+y=1.$ If $L$ also intersects the $x-$ axis, then the equation of $L$ is

1. $y-\sqrt{3}x+3\sqrt{3}+2=0$
2. $y-\sqrt{3}x-3\sqrt{3}+2=0$
3. $\sqrt{3}y-x+3+2\sqrt{3}=0$
4. $\sqrt{3}y+x-3+2\sqrt{3}=0$

Q. The coodiantes of two points on the circle $x^2+y^2-12x-16y+75=0,$ one nearest to the origin and the other farthest from it, are

1. (3, 4), (9, 12)
2. (3, 2), (9, 12)
3. (-3, 4), (9, 12)
4. (3, 4), (9, -12)

Q. If a line makes an angle of ${30}^{\circ }$ with the positive direction of $y-$ axis measured in anticlockwise direction, then the slope of the line is

1. $\sqrt{3}$
2. $-\sqrt{3}$
3. $\frac{1}{\sqrt{3}}$
4. $-\frac{1}{\sqrt{3}}$

Q. The slope of the line passing through points $(3,2)$ and $(2,5)$ is

1. $\frac{1}{3}$
2. $-3$
3. $-\frac{1}{3}$
4. $3$

Q. Circumcentre of a triangle, which is formed by the lines y=x, y=2x and y= 3x+4 is

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