The lengths of two opposite edges of a tetrahedron are \(a, b\), the angle between them is \(\theta\), and the shortest distance between them is \(d\); prove that the volume is \(\frac{1}{6}abd\sin\theta\).
A circle has double contact with an ellipse. From any point \(P\) of the ellipse \(PT\) is drawn to touch the circle at \(T\), and \(PN\) is the perpendicular to the chord of contact. Prove that \(\dfrac{PT}{PN}\) is constant.
Find the equation of the bisectors of the angles between the lines \[ ax^2+2hxy+by^2=0. \] The \(x\)-axis is reflected in each line of the pair \(ax^2+2hxy+by^2=0\). Prove that the equation of the reflexions is \[ 4abx^2+4h(a-b)xy+\{(a+b)^2-4h^2\}y^2=0. \]
Find the equation of the parabola whose focus is the origin, and whose directrix is \(x\cos\alpha+y\sin\alpha=p\). Prove that the length of the common chord of two equal parabolas having a common focus and axes inclined at an angle \(\phi\), is \(\csc^2\frac{\phi}{2}\) times the latus rectum.
Prove that there are eight normals to \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \] which touch \[ \frac{x^2}{c^2}+\frac{y^2}{d^2}=1; \] and that they are real if \(a^2-b^2>ac+bd\).
A circle cuts the rectangular hyperbola \(xy=a^2\) in the points \(A(x_1, y_1), B(x_2, y_2)\), and \(C(x_3, y_3)\). Prove that their fourth point of intersection is \[ \left( \frac{a^4}{x_1 x_2 x_3}, \frac{a^4}{y_1 y_2 y_3} \right). \] Prove also that the orthocentre of the triangle \(ABC\) is \[ \left( -\frac{a^4}{x_1 x_2 x_3}, -\frac{a^4}{y_1 y_2 y_3} \right), \] and that it lies on the hyperbola.
Prove that the equation \[ y=x+\cfrac{c^2}{x+\cfrac{c^2}{x+\dots}} \text{ to infinity} \] represents a portion of the hyperbola \[ y^2-xy=c^2. \] Sketch the curve and indicate the portion on the sketch.
A fixed point \(O\) is taken on the circumcircle of a triangle \(ABC\), and a variable point \(X\) is taken on \(BC\); the circle \(OBX\) cuts \(AB\) in \(Z\), and the circle \(OCX\) cuts \(AC\) in \(Y\). Prove that
The lines joining the vertices of a triangle \(ABC\) to any point \(O\) cut the opposite sides in \(P,Q,R\). Prove that \[ BP \cdot CQ \cdot AR = PC \cdot QA \cdot RB. \] Prove also that, if \(PQ, PR\) cut the line drawn through \(A\) parallel to \(BC\) in \(Q', R'\), then \(A\) is the mid-point of \(Q'R'\).
Shew how to construct the radical axis of two circles which do not intersect in real points. The tangents to a circle at \(P\) and \(Q\) meet in \(O\), and \(T\) is any point on the line which bisects \(OP\) and \(OQ\). Prove that the tangents from \(T\) to the circle are equal to \(TO\).