Prove that the locus of the centre of a circle, which touches two given circles in a plane, consists of two hyperbolas, two ellipses or an ellipse and a hyperbola according as one given circle is external to, internal to or cuts the other.
Three lines in space do not intersect and are not all parallel to the same plane: prove that they are three edges of a parallelepiped and that one line can be drawn to intersect the three lines so that the intercept on it between a definite pair of the lines is bisected by the third line.
Prove that the rectangles contained by the segments of any two intersecting chords of a conic are to one another as the squares of the parallel tangents taken between their intersection and their points of contact. Two chords of an ellipse \(Pp, Qq\) are at right angles and are normal to the ellipse at \(P\) and \(Q\) respectively: show that \(pq\) is parallel to \(PQ\).
Two circles lie in different planes: prove that in general four circles can be drawn to touch both circles. What is the exceptional case in which an infinite number of such circles can be drawn?
Prove that a plane section of a circular cone is a conic section as defined by focus and directrix. Prove that through any point within the cone two sections can be drawn of which the point is a focus; show also that one of these sections is an ellipse and the other an ellipse, parabola or hyperbola according as the point is within, on or without another circular cone with the same vertex and axis.
Find the respective conditions that the line \(lx+my+n=0\) (1) touches, (2) is normal to the parabola, \(y^2-4ax=0\). Two parabolas touch one another at a common vertex: prove that a chord of one parabola, which lies along a tangent to the second, is divided in a constant ratio by the common axis.
Deduce the equation \(x^2/a^2+y^2/b^2=1\) of an ellipse from the definition that it is the locus of a point \(P\), such that \(SP+PH\) is constant, where \(S\) and \(H\) are the foci. Prove that the equation of the locus of the centre of the circle inscribed in the triangle \(SPH\) is \[ (1-e)x^2 + (1+e)y^2 = a^2 e^2 (1-e). \]
Prove that the line \(lx+my+n=0\) touches the conic \(Ax^2+2Hxy+By^2=1\), provided \(Am^2 - 2Hlm + Bl^2 = (AB-H^2)n^2\). Prove that the two conics, \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{and} \quad -\frac{x^2}{a^2} - \frac{y^2}{b^2} + 2\mu\frac{xy}{ab} = (1+\mu^2)\frac{a^2+b^2}{a^2-b^2}, \] are such that any common tangent terminated by the points of contact subtends a right angle at the common centre.
Show that the conics which touch four given straight lines have their centres on a straight line. A family of conics has one focus and two tangents given: prove that the auxiliary circles and the director circles form two coaxal systems of circles, of which the respective radical axes are parallel.
Prove that two conics have four common points and four common tangents, and deduce that the relation between \(r\) and \(p\) for any conic, where \(r\) is the distance of a point on the conic from a chosen origin and \(p\) the perpendicular from the origin on the tangent, is of the fourth degree in \(r\) and in \(p\). In the case of the parabola \(y^2-4ax=0\) with the new origin at \(x=a+h, y=0\), prove that the \(p\) and \(r\) equation is \((aR-hP)^2=P^3 R\), where \(R=r^2-4ah, P=p^2-4ah\).