S L Loney Solutions for Chapter: The Hyperbola, Exercise 2: EXAMPLES XXXVII

Tangents are drawn to a hyperbola from any point on one of the branches of the conjugate hyperbola. Show that their chord of contact will touch the other branch of the conjugate hyperbola.

A straight line is drawn parallel to the conjugate axis of a hyperbola to meet it and the conjugate hyperbola in the points $P$ and $Q$ respectively. Show that the tangents at $P$ and $Q$ meet on the curve $\frac{y^4}{b^4}\left(\frac{y^2}{b^2}-\frac{x^2}{a^2}\right)=\frac{4x^2}{a^2}$ and that the normals meet on the axis of $\mathrm{x}$.

From a point $G$ on the transverse axis $GL$ is drawn perpendicular to the asymptote, and $GP$ a normal to the curve at $P$. Prove that $LP$ is parallel to the conjugate axis.

Find the asymptotes of the curve $2x^2+5xy+2y^2+4x+5y=0$, and find the general equation of all hyperbolas having the same asymptotes.

Find the equation of the hyperbola, whose asymptotes are the straight lines $x+2y+3=0$,and $3x+4y+5=0$, and which passes through the point $\left(1,-1\right)$. Also write down the equation of the conjugate hyperbola.

In a rectangular hyperbola, prove that $CP$ and $CD$ are equal, and are inclined to the axis at angles which are complementary.

$C$ is the centre of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and the tangent at any point $P$ meets the asymptotes in the points $Q$ and $R$. Prove that the equation to the locus of the centre of the circle circumscribing the triangle $CQR$ is $4\left(a^2{x}^2-b^2{y}^2\right)={\left(a^2+b^2\right)}^2$.

A series of hyperbolas are drawn having a common transverse axis of length $2a$. Prove that the locus of a point $P$ on each hyperbola, such that its distance from the transverse axis is equal to its distance from an asymptote, is the curve ${\left(x^2-y^2\right)}^2=4x^2\left(x^2-a^2\right)$.