M L Aggarwal Solutions for Chapter: Circles, Exercise 7: CHAPTER TEST

Find the equations of circles which passes through the points $P\left(1,0\right), Q\left(3,0\right)$ and $R\left(0,2\right)$. Find also the coordinates of the other end of the diameter through $Q$.

Find the value of $p$ so that the points $\left(1,1\right), \left(2,-1\right), \left(3,-2\right)$ and $\left(12,p\right)$ are concyclic.

Find the centre and the radius of the circle $x^2+y^2-4x+6y=3.$ Given that the point $A$, outside the circle, has coordinates $\left(a,b\right)$ where $a$ and $b$ are both positive and that the tangents drawn from $A$ to the circle are parallel to the two axes respectively, find the values of $a$ and $b$.

Show that the circle $x^2+y^2-(3p+4)x-(p-2)y+10p=0$ passes through the point $\left(3,1\right)$ for all values of $p$. If $p$ varies, find the equation of the locus of the centre of the circle.

Show that the circle $x^2+y^2-(3p+4)x-(p-2)y+10p=0$ passes through the point $\left(3,1\right)$ for all values of $p$. If $p$ varies, find the value of $p$ for which the line $x=3$ is a tangent to the circle.

Find the equation of the circle which has extremities of a diameter the origin and the point $\left(2,-4\right)$. Find the also the equations of tangents to the circle which are parallel to this diameter.

A circle touches the $y$-axis at the point $\left(0,4\right)$ and passes through the point $\left(2,0\right)$. Find the equation of the circle.

If the two circles $(x-1{)}^2+(y-3{)}^2=r^2$ and $x^2+y^2-8x+2y+8=0$ intersect in two distinct points, then prove that $2<r<8.$