A point \(P\) moves in a plane so that the ratio of its distances from two fixed points \(A\) and \(B\) in the plane is a constant different from 1. Prove the locus of \(P\) to be a circle, \(S\). A line through \(A\) cuts \(S\) in \(Q, R\). Prove that \(QB, RB\) are equally inclined to \(AB\).
The cardioid whose equation in polar coordinates is \[ r = a(1+\cos\theta) \] is inverted with respect to a circle with centre at the origin. Prove, from the focus-directrix definition of a parabola, that the inverse figure is a parabola with focus at the origin. A variable circle through the origin touches the cardioid at a further point; what is the locus of its centre?
The circumscribing sphere of a tetrahedron \(A_1A_2A_3A_4\) has centre \(Q\); \(O_1\) is the circumcentre of \(A_2A_3A_4\) and \(O_2, O_3, O_4\) are similarly defined. Prove that \(QO_1\) is perpendicular to the plane \(A_2A_3A_4\). Prove that, if \(\angle A_3A_1A_4 = \angle A_3A_2A_4\), the plane \(QA_3A_4\) bisects \(O_1O_2\) perpendicularly. If also \(\angle A_1A_3A_2 = \angle A_1A_4A_2\), prove that the line through \(Q\) that intersects \(A_1A_2\) and \(A_3A_4\) is perpendicular to (without necessarily intersecting) \(O_1O_2\) and \(O_3O_4\).
From a variable point \(P\) on the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \quad (a>b) \] tangents are drawn to the circle \[ x^2+y^2=b^2. \] Through the origin \(O\) a line is drawn perpendicular to \(OP\) to cut these tangents in \(T_1, T_2\). Find the locus of \(T_1, T_2\).
A line cuts the asymptotes \(l_1, l_2\) of a hyperbola in two distinct points \(P_1, P_2\). The line through \(P_1\) parallel to \(l_2\) cuts \(S\) in \(Q_1\) and the line through \(P_2\) parallel to \(l_1\) cuts \(S\) in \(Q_2\). Prove that \(Q_1Q_2\) is parallel to \(P_1P_2\). Prove that, if the tangents to \(S\) at \(Q_1, Q_2\) meet on \(P_1P_2\), then \(P_1P_2\) envelops another hyperbola with \(l_1, l_2\) as asymptotes.
Prove Desargues' theorem that if two triangles in a plane are in perspective the intersections of their corresponding sides are collinear. \(ABC, A'B'C'\) are two coplanar triangles which are in perspective, and \(BC'\) meets \(B'C\) in \(P\). \(CA'\) meets \(C'A\) in \(Q\), and \(AB'\) meets \(A'B\) in \(R\). Prove that the triangle \(PQR\) is in perspective with \(ABC\).
A triangle \(ABC\) is inscribed in a conic \(S\). The tangents to \(S\) at \(B\) and \(C\) meet in \(A'\), and \(B', C'\) are similarly defined. \(L\) is an arbitrary point of the line \(BC\); \(C'L\) meet in \(M\) and \(B'L\) meet in \(N\). Prove that \(M, N, A'\) are collinear and that the triangle \(LMN\) is self-polar with respect to \(S\).
The cartesian coordinates of the points of a hyperbola are expressed in the parametric form \((p\theta+q\theta^{-1}+r, p'\theta+q'\theta^{-1}+r')\), where \(p,q,r,p',q',r'\) are fixed and \(pq' \neq p'q\). Find (i) the equation of the tangent to the hyperbola at a general point \(\theta\), (ii) the equations of the asymptotes, (iii) the coordinates of the centre.
Find the necessary and sufficient condition that the two pairs of lines \[ ax^2+2hxy+by^2=0, \quad a'x^2+2h'xy+b'y^2=0 \] should be harmonically conjugate. Prove that, if \(\lambda\mu\nu=1\), any tangent to one of the conics \[ x^2+2\lambda y=0, \quad y^2+2\mu x=0, \quad 2\nu xy+1=0, \] cuts the other two conics in harmonically conjugate pairs of points.
A series of circles is drawn with given centre \(O\). Show that the mid-points of their chords of intersection with a given central conic \(S\) lie on a rectangular hyperbola \(\Gamma\), whose asymptotes are parallel to the axis of \(S\) and which passes through \(O\) and the centre of \(S\). Show further that the same rectangular hyperbola \(\Gamma\) is obtained if \(S\) is replaced by any conic \(S'\) which passes through the common points of \(S\) and any circle with centre \(O\).