M L Aggarwal Solutions for Exercise 4: Exercise 19.4

The centre of a circle is $C\left(2\alpha -1,3\alpha +1\right)$ and it passes through the point $A\left(-3,-1\right)$. If a diameter of the circle is of length $20 \text{units}$, find the value(s) of $\alpha$.

Using distance formula, show that the points $A\left(3,1\right),B\left(6,4\right) \text{and} C\left(8,6\right)$ are collinear.

Check whether the points $\left(5,-2\right),\left(6,4\right) \text{and} \left(7,-2\right)$ are the vertices of an isosceles triangle.

Name the type of triangle formed by the points $A\left(-5,6\right),B\left(-4,-2\right) \text{and} C\left(7,5\right)$

The points $A (0, 3), B (- 2, a)$ and $C(- 1, 4)$ are the vertices of a right angled triangle at A, find the value of a.

Show that the points $\left(0,-1\right),\left(-2,3\right),\left(6,7\right) \text{and} \left(8,3\right)$, taken in order, are the vertices of a rectangle. Also find its area.

If $P\left(2,-1\right),Q\left(3,4\right),R\left(-2,3\right) \text{and} S\left(-3,-2\right)$ be four points in a plane, show that $PQRS$ is a rhombus but not a square. Find the area of the rhombus.

Find the coordinates of the circumcentre of the triangle whose vertices are $\left(8,6\right),\left(8,-2\right) \text{and} \left(2,-2\right)$. Also, find its circumradius.