\(OP, OQ\) are two variable lines at right angles through a fixed point \(O\). Prove that the join of the mid-points of the chords intercepted on \(OP, OQ\) by a fixed conic passes through a fixed point.
Find the equation of the normal to the parabola \(y^2=4ax\) at the point \((at^2, 2at)\). If the normals at the three points where \(t\) has the values \(t_1, t_2, t_3\) form an equilateral triangle, prove that \begin{align*} t_1+t_2+t_3+3t_1t_2t_3 &= 0, \\ t_1t_2+t_2t_3+t_3t_1+3 &= 0. \end{align*} Show that the locus of the centre of the equilateral triangle is the parabola \[ 3y^2 = 2a(x-5a). \]
Prove that two confocal conics cut everywhere at right angles. Prove that, if the two conics \(ax^2+by^2=1\) and \(\alpha x^2+\beta y^2+2\gamma xy=1\) cut at right angles at all four points of intersection, then either they are confocal or else \[ \frac{\alpha-a}{a\alpha} = \frac{\beta-b}{b\beta} = \frac{2}{a+b}. \]
Prove that the conics, which have a given triangle \(XYZ\) as a self-polar triangle, and for which two given lines \(OP, OQ\) are conjugate, form a tangential pencil. Prove that the four common tangents of the conics are the four chords of contact of \(OP, OQ\) with the conics which touch \(OP\) and \(OQ\) and pass through \(X, Y, Z\).
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 \(ABC\).