Embibe Experts Solutions for Chapter: Circle, Exercise 4: Exercise-4

Show that if one of the circle $x^2+y^2+2gx+c=0$ and $x^2+y^2+2g_{1}x+c=0$ lies within the other, then $g{g}_1$ and $C$ are both positive.

Let $ABCD$ is a rectangle. The incircle of $∆ABD$ touches $BD$ at $E$. Incircle of $\Delta CBD$ toches $BD$ at $F$. If $AB=8$ units, and $BC=6$ units, then find the length of $EF$.

Let circles $S_1$ and $S_2$ of radii $r_1$ and $r_2$ respectively $\left(r_1>r_2\right)$ touches each other externally. Circle $S$ of radius $r$ touches $S_1$ and $S_2$ externally and also their direct common tangent. Prove that the triangle formed by joining centre of $S_1, S_2$ and $S$ is obtuse angled triangle.

Circles are drawn passing through the origin $O$ to intersect the coordinate axes at points $P \& Q$ such that $m·OP+n·OQ$ is a constant. Show that the circles pass through a fixed point

The curves whose equations are
$S=a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ 

$S^{\text{'}}=a^{\text{'}}{x}^2+2h^{\text{'}}xy+b^{\text{'}}{y}^2+2g^{\text{'}}x+2f^{\text{'}}y+c^{\text{'}}=0$  intersect in four concyclic points then find relation in $a, b, h, a^{\text{'}}, b^{\text{'}}, h^{\text{'}}.$

A circle of constant radius $\text{'}r\text{'}$ passes through the origin $O$ and cuts the axes of coordinates in points $P$ and $Q$, then find the equation of the locus of the foot of the perpendicular from $O$ to $PQ$.

The ends $A, B$ of a fixed straight line of length $\text{'}a\text{'}$ and ends $A^{\text{'}}$ and $B^{\text{'}}$ of another fixed straight line of length $\text{'}b\text{'}$ slide upon the $x$-axis and $y$-axis (one end on axis of $x$ and  the other on axis of $y$ ). Find the locus of the centre of the circle passing through $A, B, A^{\text{'}}$ and $B^{\text{'}}$

Let $P$ be a point from where perpendicular tangents are drawn to the circle $2x^2+2y^2-a^2=0$. Let a line from $P$perpendicular to $OP$ is drawn which intersect hyperbola $x^2-y^2=a^2$ at $Q$ and $R$. Find number of all possible positions of $P$ such that product of ordinates of points $Q$ and $R$ is $\frac{3}{2}{a}^2$.