Prove that any chord of a rectangular hyperbola subtends, at the ends of any diameter, angles which are equal or supplementary. A conic is inscribed in a parallelogram. Prove that its foci lie on a rectangular hyperbola which passes through the angular points of the parallelogram. Find its asymptotes and axes.
Prove that a conic and any point in its plane can be projected into a circle and its centre respectively, and that any two conics can be simultaneously projected into circles. How must two conics be related so that they can be projected into two circles such that the centre of one circle shall lie on the circumference of the other?
\(A\) is a fixed point on a sphere, \(P\) a variable point on it. \(AP\) is produced to \(Q\) so that \(PQ\) is of constant length. Prove that the plane through \(Q\) perpendicular to \(PQ\) envelopes a sphere.
Find the equation of the polar of a point with respect to the circle \[ x^2+y^2+2gx+2fy+c=0. \] Prove that the locus of the poles of a given line with respect to a coaxal system of circles is a hyperbola whose asymptotes are respectively perpendicular to the given line and parallel to the common radical axis of the circles.
Find the equation of the pair of tangents from the point \(P(x', y')\) to the parabola \(y^2=4ax\). If they intercept a constant length \(l\) on the directrix, prove that the locus of \(P\) is \[ (y^2-4ax)(x+a)^2=l^2x^2. \]
Define the eccentric angle at a point of an ellipse. Find the equation of the tangent to the ellipse \[ x^2/a^2+y^2/b^2=1, \] at the point whose eccentric angle is \(\phi\). A circle is described through the centre and one focus to touch the ellipse. Prove that the eccentric angle of the point of contact is \[ \cos^{-1}[\{(1+4e^2)^{\frac{1}{2}}-1\}/2e]. \]
Prove that the locus given by \[ x=at^2+2bt+a', \quad y=a't^2+2b't+a, \] where \(t\) is a variable parameter is a parabola. If \(t_1, t_2\) are the parameters of the extremities of any chord parallel to \(y=mx\), prove that \[ t_1+t_2 = 2(b'-mb)/(am-a'). \]
Define conjugate lines with respect to a conic. Prove that \[ lx+my+n=0, \quad \text{and} \quad l'x+m'y+n'=0 \] are conjugate with respect to \[ ax^2+2hxy+by^2+2gx+2fy+c=0, \] if \[ All'+Bmm'+Cnn'+F(mn'+m'n)+G(nl'+n'l)+H(lm'+l'm)=0, \] where \(A, B, \dots\) are the minors of \(a, b, \dots\) in \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}. \]
Show how to construct the fourth harmonic of a given point with respect to two given points in the same line with it. Through a given point \(O\), draw a straight line to cut the sides of a given triangle in points \(A', B', C'\), so that \((OA'B'C')\) may be a harmonic range.
Prove that if two chords of a circle are perpendicular the tangents at their ends form a quadrilateral inscribed in a circle; and that if the intersection of the chords is a fixed point the circumcircle remains fixed when the chords revolve. Also reciprocate this theorem with respect to a circle having the fixed point as centre.