S L Loney Solutions for Chapter: The Hyperbola, Exercise 3: EXAMPLES XXXVIII

Prove that the locus of the poles of all normal chord of the rectangular hyperbola $xy=c^2$, is the curve ${\left(x^2-y^2\right)}^2+4c^{2}xy=0$.

Prove that the triangle can be inscribed in the hyperbola $xy=c^2$, whose sides touch the parabola $y^2=4ax$.

A point moves on the given straight line $y=mx$, prove that the locus of the foot of the perpendicular let fall from the point upon its polar with respect to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, is a rectangular hyperbola one of whose asymptote is the diameter of the ellipse which is conjugate to the given straight line.

If from a fixed point on a rectangular hyperbola, perpendiculars are let fall on any two conjugate diameters, prove that the straight line joining the feet of these perpendiculars has a constant direction.

A quadrilateral circumscribes a hyperbola, prove that the straight line joining the middle points of its diagonals passing through the center of the curve.

$A,B,C$ and $D$ are the points of intersection of a circle and a rectangular hyperbola. If $AB$ passes through the center of the hyperbola, prove that $CD$ passes through the center of the circle.

If a circle and rectangular hyperbola meet in four points $P, Q, R$ and $S$, show that the orthocenter of the triangles $QRS, RSP, SPQ$ and $PQR$, also lie on a circle. Prove also that the tangents to the hyperbola at $R$ and $S$, meet in a point which lies on the diameter of the hyperbola which is at right angles to $PQ.$

A series of hyperbolas is drawn, for an asymptote, the principal axes on ellipse. Prove that the common chord of the hyperbola and the ellipse are all parallel to one of the conjugate diameters of the ellipse.