Given a self-conjugate triangle with respect (i) to a circle, construct the circle; (ii) to a rectangular hyperbola, show that the locus of the centre of the hyperbola is the circumcircle of the triangle.
Prove that the polar reciprocal of a circle with respect to another circle is a conic and find the positions of the centre, foci and asymptotes of the conic. Reciprocate the theorem that the angle in the segment of a circle is constant.
\(PSQ\) is a focal chord of a conic whose focus \(S\) lies between \(P\) and \(Q\). The tangents at \(P\) and \(Q\) meet in \(T\), prove that \(TS\) is perpendicular to \(PQ\). Show that the conic is an ellipse, parabola or hyperbola according as the angle \(PTQ\) is less than, equal to or greater than a right angle.
If \(S=0, S'=0\) denote circles, prove that \(S+\lambda S'=0\) represents a system of coaxal circles. Find the limiting points of the coaxal system of circles determined by \[ x^2+y^2-2x+2y-8=0, \quad x^2+y^2-4x-12y+39=0. \]
Find the equation of the normal at any point on \(y^2=4ax\). From any point on this parabola, two normal chords of lengths \(n, n'\) are drawn to the curve, prove that the radius of curvature at the given point is equal to \(nn'/2a\).
Find the equation of the pair of tangents from \((x',y')\) to \(x^2/a^2+y^2/b^2=1\). If the product of the tangents from a point \(P\) to this ellipse is constant and equal to \(c^2\), prove that the locus of \(P\) is \[ \{(a^2y^2+b^2x^2)c^2/(a^2y^2+b^2x^2-a^2b^2)\}^2 = (x^2+y^2)^2-2(a^2-b^2)(x^2-y^2)+(a^2-b^2)^2. \]
Prove that the line \(lx+my+n=0\) touches a conic if \[ Al^2+Bm^2+Cn^2+2Fmn+2Gnl+2Hlm=0. \] Prove that the conic is a parabola if \(C=0\) and in this case find the coordinates of its focus and the equation of its axis.
Find the locus of a point which moves so that its distances from two fixed points are in a constant ratio. A square \(ABCD\) being given, prove that, if \(E\) is that point on \(AD\) produced at which the ratio \(BE:CE\) has its greatest value, then \(AD^2 = EA \cdot ED\).
Segments \(PP', QQ', RR', SS'\) of a straight line subtend at a point equal angles in the same sense. Prove, ab initio, that the cross-ratios of \(PQRS\) and \(P'Q'R'S'\) are equal. Two variable lines \(POP'\) and \(QOQ'\) through a fixed point \(O\) meet two fixed lines in points in \(P, P'\) and \(Q, Q'\). If \(PQ\) subtends a constant angle at a fixed point, shew that \(P'Q'\) subtends a constant angle at another fixed point, and that the angle is the same.
Prove that if a sphere passes through the eight vertices of a parallelepiped the parallelepiped must be rectangular. Mutually perpendicular chords \(AP, AQ, AR\) are drawn through any point \(A\) of a sphere. Shew that the plane \(PQR\) cuts the diameter through \(A\) at a point of trisection.