Circle Inscribing a Triangle Fromed by 3 Given Lines

Q. In $\Delta ABC$, the coordinates of vertex $A$ is $(4,-1)$, and the lines $x-y-1=0$ and $2x-y=3$ are the internal bisectors of the angles $B$ and $C$ respectively. If the radius of the incircle of the triangle $ABC$ is $r$ then the value of [r] is
(where [.] denotes the greater integer function)

Q. The equation of the circle circumscribing the triangle formed by the lines $x+y=6,2x+y=4$ and $x+2y=5$ is

1. $x^2+y^2+17x+19y+50=0$
2. $x^2+y^2-19x-17y+50=0$
3. $x^2+y^2-17x-19y+50=0$
4. $x^2+y^2+19x-17y+50=0$

Q. An isosceles triangle $ABC$ is inscribed in $x^2+y^2=a^2$ with the vertex $A$ at $(a,0)$ and the base angle $B$ and $C$ each equal to ${75}^{\circ }$, then coordinates of an endpoint of the base are

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

Q. The equation of the circle circumscribing the triangle formed by the lines $x+y=6,2x+y=4$ and $x+2y=5$ is

1. $x^2+y^2-17x-19y+50=0$
2. $x^2+y^2+17x+19y+50=0$
3. $x^2+y^2-19x-17y+50=0$
4. $x^2+y^2+19x-17y+50=0$

Q.

The lines 2x-3y=5 and 3x-4y=7 are the diameters of a circle of area 154 square units. The equation of the circle is

1. 

2. 

3. 

4. 

Q. Suppose $z_1,z_2,z_3$ are the vertices of an equilateral triangle inscribed in the circle $|z|=2$. If $z_1=1+i\sqrt{3}$. then values of $z_{3}and{z}_2$ are respectively

1. 
2. 
3. 
4. 

Q.

Find the equation of the circle circumscribing the triangle formed by the lines $L_1=0,L_2=0$and $L_3=0$

1. $L_1+\lambda L_2+\mu L_3=0$

2. $L_1{L}_2+\lambda L_2{L}_3+\mu L_3{L}_1=0$

3. $L_1{L}_2{L}_3=0$

4. $L_1^2+\lambda L_2^2+\mu L_3^2=0$

Q. The equation of the circle circumscribing the triangle formed by the lines $x+y=6,2x+y=4$ and $x+2y=5$ is

1. $x^2+y^2-17x-19y+50=0$
2. $x^2+y^2+17x+19y+50=0$
3. $x^2+y^2-19x-17y+50=0$
4. $x^2+y^2+19x-17y+50=0$

Q. The equation of the circle circumscribing the triangle formed by the lines $x+y=6,2x+y=4$ and $x+2y=5$ is

1. $x^2+y^2-17x-19y+50=0$
2. $x^2+y^2+17x+19y+50=0$
3. $x^2+y^2-19x-17y+50=0$
4. $x^2+y^2+19x-17y+50=0$

Q. If $\alpha ,\beta ,\gamma$ are the real roots of the equation $x^3-3p{x}^2+3qx-1=0$, then the centroid of the triangle with vertices $(\alpha ,\frac{1}{\alpha })$, $(\beta ,\frac{1}{\beta })$ and $(\gamma ,\frac{1}{\gamma })$ is at the point

1. $(p,q)$
2. $(\frac{p}{3},\frac{q}{3})$
3. $(p+q,p-q)$
4. $(3p,3q)$

Q. The equation of the circumcircle of the triangle formed by the lines x = 0, y = 0, 2x + 3y = 5 is

1. $x^2+y^2+2x+3y-5=0$
2. $6(x^2+y^2)-5(3x+2y)=0$
3. $x^2+y^2-2x-3y+5=0$
4. $6(x^2+y^2)+5(3x+2y)=0$

Q. Tangents are drawn from any point on the circle $x^2+y^2=R^2$ to the circle $x^2+y^2=r^2$. If the line joining the points of intersection of these tangents with the first circle also touch the second, then R equals

1. 
2. 
3. 
4. 

Q. The equation of the circumcircle of the triangle formed by the lines x = 0, y = 0, 2x + 3y = 5 is

1. $6(x^2+y^2)-5(3x+2y)=0$
2. $x^2+y^2-2x-3y+5=0$
3. $6(x^2+y^2)+5(3x+2y)=0$
4. $x^2+y^2+2x+3y-5=0$

Q. An isosceles triangle $ABC$ is inscribed in $x^2+y^2=a^2$ with the vertex $A$ at $(a,0)$ and the base angle $B$ and $C$ each equal to ${75}^{\circ }$, then coordinates of an endpoint of the base are

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

Q. The equation of the circle circumscribing the triangle formed by the lines $x+y=6,2x+y=4$ and $x+2y=5$ is

1. $x^2+y^2-17x-19y+50=0$
2. $x^2+y^2+17x+19y+50=0$
3. $x^2+y^2-19x-17y+50=0$
4. $x^2+y^2+19x-17y+50=0$

Q.

Find the equation of the circle circumscribing the triangle formed by the lines $x+y=0,2x+y=4$ and $x+2y=5$

1. x² + y² - 17x - 19y + 50 = 0

2. x² + y² + 17x + 19y + 50 = 0

3. x² + y² + 17x - 11y + 30 = 0

4. x² + y² - 17x - 11y + 30 = 0

Q.

Find the equation of the circle circumscribing the triangle formed by the lines $L_1=0,L_2=0$and $L_3=0$

1. 

2. 

3. 

4.