S L Loney Solutions for Chapter: The Circle, Exercise 2: EXAMPLES XVIII

Find the equation of the circle which has its center at the point $\left(3, 4\right)$ and touches the straight line, $5x+12y=1$.

Find the equation of the circle which touches the axes of the coordinates and also the line $\frac{x}{a}+\frac{y}{b}=1$, the center being in the positive quadrant.

Find the equation of the circle which has its center at the point $(1, -3)$ and touches the straight line $2x-y-4=0$.

Find the general equation of a circle referred to two perpendicular tangents as the axes.

Find the equation of a circle of radius $r$, which touches the axis of $y$ at a point at distance $h$ from the origin, the center of the circle being in the positive quadrant. Also, prove that the equation of the other tangent which passes through the origin is, $\left(r^2-h^2\right)x+2rhy=0$.

Find the equation of the circle whose center is at the point $(\mathrm{\alpha }, \mathrm{\beta })$ and which passes through the origin, and prove that the equation of the tangent at the origin is, $\mathrm{\alpha x}+\mathrm{\beta y}=0$.

Two circles are drawn through the points $(a, 5a)$ and $(4a, a)$ to touch the axis of $y$, where $a>0$. Prove that they intersect at an angle ${\tan }^{-1}\left(\frac{40}{9}\right)$.

A circle passes through the points $(-1, 1), (0, 6)$ and $(5, 5)$. Find the points on this circle the tangents at which are parallel to the straight line joining the origin to its center.