Excentre

Q. If $P(x,y)$ divides the line segment joining $A(5,0)$ and $B(10\cos \theta ,10\sin \theta )$ internally in the ratio $2:3,$ then which of the following is (are) true?

1. Distance of $P$ from $(3,0)$ is always constant.
2. $x\in [0,3]$
3. $y\in [-4,0]$
4. $x^2+y^2-6x-7=0$

Q. The vertices of a triangle are $A(0,-6),$ $B(-6,0)$ and $C(1,1)$ respectively. Then coordinates of the excentre opposite to vertex $A$ are

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

Q. If the point $(2,a)$ divides the line segment joining the points $(-4,3)$ and $(6,3),$ then

1. the ratio is $2:3$
2. the ratio is $3:2$
3. the value of $a$ is $3$
4. the value of $a$ is $2$

Q. If the vertices of a triangle are $A(2,4),B(2,6)$ and $C(2+\sqrt{3},5)$, then which of the following is/are true?

1. Coordinates of the excentre opposite to vertex $A$ is $(2+\sqrt{3},7)$
2. Coordinates of the excentre opposite to vertex $B$ is $(2-\sqrt{3},3)$
3. Coordinates of the excentre opposite to vertex $C$ is $(2-\sqrt{3},5)$
4. The area of $△ABC$ is $\sqrt{3}\text{sq. units}$

Q. If the point $(2,a)$ divides the line segment joining the points $(-4,3)$ and $(6,3),$ then

1. the ratio is $2:3$
2. the ratio is $3:2$
3. the value of $a$ is $3$
4. the value of $a$ is $2$

Q. If $P(x,y)$ divides the line segment joining $A(5,0)$ and $B(10\cos \theta ,10\sin \theta )$ internally in the ratio $2:3,$ then which of the following is (are) true?

1. Distance of $P$ from $(3,0)$ is always constant.
2. $x\in [0,3]$
3. $y\in [-4,0]$
4. $x^2+y^2-6x-7=0$

Q. The vertices of a triangle are $A(0,-6),$ $B(-6,0)$ and $C(1,1)$ respectively. Then coordinates of the excentre opposite to vertex $A$ are

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

Q. If the vertices of a triangle are $A(2,4),B(2,6)$ and $C(2+\sqrt{3},5)$, then which of the following is/are true?

1. Coordinates of the excentre opposite to vertex $A$ is $(2+\sqrt{3},7)$
2. Coordinates of the excentre opposite to vertex $B$ is $(2-\sqrt{3},3)$
3. Coordinates of the excentre opposite to vertex $C$ is $(2-\sqrt{3},5)$
4. The area of $△ABC$ is $\sqrt{3}\text{sq. units}$