Dr. SK Goyal Solutions for Chapter: Circle, Exercise 8: EXERCISE ON LEVEL-I

Show that the equation of a straight line meeting the circle ${\text{x}}^2+{\text{y}}^2={\text{a}}^2$ in two points at equal distances $d$ from a point $\left({\text{x}}_1\text{, }{\text{y}}_1\right)$ on its circumference is ${\text{xx}}_1{+\text{yy}}_1-{\text{a}}^2+\left({\text{d}}^2/2\right)=0$ .

Show that the line $\left(x-1\right)\cos \theta +\left(y-1\right)\sin \theta =1$ touches a circle for all values of $\theta$. Find the circle.

Show that the line $3x-4y-c=0$ will meet the circle having centre at $\left(2,4\right)$ and the radius $5$ in real and distinct points if $-35<c<15$.

A rectangle $ABCD$ is inscribed in a circle with a diameter lying along the line $3y=x+10$. If $A$ and $B$ are the points $\left(-6,7\right)$ and $\left(4,7\right)$ respectively, find the area of the rectangle.

Find the equations of smallest circles which cut the circles $x^2+y^2-4x-6y+11=0$ and $x^2+y^2-10x-4y+21=0$ orthogonally.

The circle $x^2+y^2-6x-6y+9=0$ is rolled on the $x$-axis in the positive direction through one complete revolution. Find the equation of the circle in the new position.

A circle of diameter $13\mathrm{m}$ with the centre $O$, coinciding with the origin of co-ordinate axes has the diameter $AB$ on the $x$-axis ( $x$ co-ordinate of $B>0$). If the length of the chord $AC$ be $5\mathrm{m}$, find the equation of pair of lines $BC, C$ having two possible positions.

$AB$ is a diameter of a circle. $CD$ is a chord parallel to $AB$ and $2CD=AB$. The tangent at $B$ meets the line $AC$ produced at $E$. Prove that $AE=2AB$.