S L Loney Solutions for Chapter: The Circle, Exercise 1: EXAMPLES XVII

Find the equation of the circle which touches the axis of $y$ at the origin and passes through the point $\left(b, c\right), b,c>0$.

Find the equation of the circle when the circle touches the axis of $x$ at a distance of $3$ units from the origin and intercepts a distance $6$ on the axis of $y$.

Points $\left(1, 0\right)$ and $\left(2, 0\right)$ are taken on the x-axis, the axes being rectangular. On the line segment joining these points an equilateral triangle is described, whose third vertex being in the first quadrant, find the equations of the circles described on sides of triangle as diameters.

If $y=mx$ is the equation of a chord of a circle whose radius is $a$, the origin of the coordinates being one extremity of the chord and the axis of $x$ being the diameter of the circle, prove that the equation of a circle of which this chord is, has the diameter as $\left(1+m^2\right)\left(x^2+y^2\right)-2a\left(x+my\right)=0$.

Find the equation of the circle passing through the points $\left(12,43\right)$, $\left(18,39\right)$, and $\left(42,3\right)$ and prove that it also passes through the points $\left(-54,-69\right)$ and $\left(-81,-38\right)$.

Find the equation of the circle circumscribing the quadrilateral formed by the straight lines $2x+3y=2, 3x-2y=4, x+2y=3$ and $2x-y=3$, respectively.

Prove that the equation of the circle of which the points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ are the ends of a chord of a segment containing an angle $\theta$ is 

$\left(x-x_1\right)\left(x-x_2\right)+\left(y-y_1\right)\left(y-y_2\right)\pm \cot \mathrm{\theta }\left[\left(x-x_1\right)\left(y-y_2\right)-\left(x-x_2\right)\left(y-y_1\right)\right]=0$.

Find the equations of the circles in which the line joining the points $(a, b)$ and $(b, -a)$ is a chord subtending an angle $45°$ at any point on its circumference.