Two circles intersect in \(P\) and \(Q\). Draw a straight line through \(P\) so that the segments of the line cut off by the circles shall be in a given ratio.
A circle is inscribed in a right-angled triangle and another is escribed to one of the sides containing the right angle. Prove that the lines joining the points of contact with the hypotenuse and that side in each circle intersect one another at right angles, and being produced pass each through the point of contact of the other circle with the remaining side.
Prove that the inverse of a circle is a circle or a straight line, and find in each case the point into which the centre of the given circle inverts. A circle of radius \(a\) and centre \(C\) is inverted from a point \(O\) with respect to a circle of radius \(K\) into a circle of radius \(a'\) and centre \(C'\), and \(P'\) is the inverse of a point \(P\) in the same plane as the circles. Prove that \[ OP^2(a'^2-C'P'^2)(OC^2-a^2) = K^4 OP'^2(a^2-CP^2). \]
Prove the harmonic properties of a complete quadrilateral. \(ABCD\) is a quadrilateral, \(AB\) and \(CD\) meet in \(Q\), \(BC\) and \(AD\) meet in \(R\), \(AC\) and \(BD\) in \(P\). Shew that if \(PQ\) meets \(AD\) in \(S\) and \(PR\) meets \(AB\) in \(T\), then \(BS, DT\) and \(AC\) are concurrent and \(ST, BD\) and \(QR\) are concurrent.
Prove that in an ellipse \(SP.S'P = CD^2\), where \(CD\) is the semi-diameter conjugate to \(CP\). Tangents drawn to an ellipse at the extremities of two conjugate semi-diameters intersect in \(P\). Prove that the rectangle contained by the focal distances of \(P\) cannot exceed the sum of the squares of the semi-axes of the ellipse.
Prove that the equations of two circles cutting at right angles may be put in the form \[ x^2+y^2-2cx\cot\theta = c^2 \quad \text{and} \quad x^2+y^2+2cx\tan\theta = c^2. \] Prove that the locus of points at which these circles subtend equal angles is \[ x^2+y^2+2cx\tan 2\theta = c^2. \]
Find the invariants of \(ax^2+2hxy+by^2+2gx+2fy+c\) for a transformation from one set of rectangular axes to another. Prove that the transformation of rectangular axes which converts \(X^2/\alpha+Y^2/\beta\) into \(ax^2+2hxy+by^2\) will convert \(X^2/(\alpha-\theta)+Y^2/(\beta-\theta)\) into \[ \{ax^2+2hxy+by^2-\theta(ab-h^2)(x^2+y^2)\}/\{1-(a+b)\theta+(ab-h^2)\theta^2\}. \]
A circle touches a hyperbola at two points, the chord of contact being parallel to the transverse axis. Prove that the length of the tangent to the circle from any point of the hyperbola is to the distance of the point from the chord of contact as \(e:1\), where \(e\) is the eccentricity of the conjugate hyperbola.
Interpret the equation \(S-\alpha T=0\), where \(S=0\) is a conic, \(T=0\) is the tangent to the conic at a point \(P\) on it and \(\alpha=0\) the equation of a straight line through \(P\). Find the equation of the circle of curvature at the point \(P(ct, c/t)\) on the rectangular hyperbola \(xy=c^2\). If \(\rho\) is the radius of curvature, \(n\) the length of the normal chord of the hyperbola at \(P\), and \(l\) the length of the tangent from the centre of the hyperbola to the circle of curvature at \(P\), prove that \[ n=2\rho=l^2/(3\sqrt{3}c^2). \]
Find the condition that the line \(lx+my+nz=0\) should touch the conic \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0. \] Prove that four conics can be drawn through the vertices of the triangle of reference to touch the two lines \(lx+my+nz=0, l'x+m'y+n'z=0\), and shew that the equations of the chords of contact of the conics with these lines in the four possible cases are \[ x(ll')^{1/2} \pm y(mm')^{1/2} \pm z(nn')^{1/2} = 0. \]