Prove that the points in which a straight line meets a circle are harmonically conjugate with respect to any point on that line and the point of intersection of the line with the polar of the point. Prove that, if four points of intersection of four straight lines lie on a circle, the centre of the circle is the orthocentre of the triangle formed by the diagonals of the quadrilateral formed by the lines.
Prove that in an ellipse \(SP.S'P=CD^2\), where \(S\) and \(S'\) are the foci and \(CD\) is the diameter conjugate to \(CP\). Tangents at the extremities of conjugate diameters \(CP, CD\) of an ellipse meet in \(T\), and \(SD\) is produced to \(Q\) so that \(SD=DQ\). Shew that the triangle \(SQT\) is similar to the triangle \(STP\).
Prove that any chord of a rectangular hyperbola subtends equal or supplementary angles at the extremities of any diameter. \(AA'\) is any diameter of a rectangular hyperbola, \(PP'\) is a chord perpendicular to \(AA'\); shew that the circum-circle of the triangle \(PP'A\) touches the hyperbola at \(A\).
Prove that the equations of any two circles can be put in the form \[ x^2+y^2+2kx+c=0 \quad \text{and} \quad x^2+y^2+2k'x+c=0. \] Find the condition that one circle should lie within the other. Find the equation of the coaxal system of circles whose limiting points are \((0,0)\) and \((a,b)\).
Find the equation of the tangent at the point on the ellipse \(x^2/a^2+y^2/b^2=1\) whose eccentric angle is \(\phi\). If the circle through the centre \(C\) and focus \(S\) touch the ellipse at \(P\), prove that \(CN:CS = PN^2:BC^2\), where \(PN\) is the ordinate at \(P\).
Shew that the semi-axes of the conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\) are the roots of the equation \(C^2r^4+C\Delta(a+b)r^2+\Delta^2=0\) where \(C\) is \(ab-h^2\) and \(\Delta\) is \[ \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}. \] Prove that when the conic is referred to its asymptotes as axes of coordinates its equation is \[ 4Cxy-(a-b+4h^2)^{\frac{1}{2}}\Delta=0. \]
Find the values of \(\cos 15^\circ\) and \(\sin 18^\circ\). If \(\cos(\alpha+\beta+\gamma)+\cos(\beta+\gamma-\alpha)+\cos(\gamma+\alpha-\beta)+\cos(\alpha+\beta-\gamma)=0\), prove that one of the angles \(\alpha, \beta, \gamma\) must be an odd multiple of a right angle. Prove also that if \[ \sec(\alpha+\beta+\gamma)+\sec(\beta+\gamma-\alpha)+\sec(\gamma+\alpha-\beta)+\sec(\alpha+\beta-\gamma)=0, \] either \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=2\) or else one of the angles \(\alpha, \beta, \gamma\) is an odd multiple of a right angle.
Expand \(\cos n\theta\) in a series of ascending powers of \(\cos\theta\). Prove that \(\sum_{r=0}^{r=\frac{n-1}{2}} \sec^2\left(\alpha+\frac{2r\pi}{n}\right)\) is equal to \(n^2\sec^2 n\alpha\) when \(n\) is odd.
Prove the formulae in the case of a triangle:
If \(\alpha, \beta\) denote the roots of a given quadratic equation \(Ax^2+Bx+C=0\), find the quadratic of which the roots are \(\frac{a\alpha^2+b\alpha+c}{a'\alpha^2+b'\alpha+c'}\) and \(\frac{a\beta^2+b\beta+c}{a'\beta^2+b'\beta+c'}\). Prove that, if \(x\) be restricted to be real, \(\frac{kx^2+kx+1}{x^2+kx+k}\) can have all values in case \(k\) is negative and not numerically less than \(\frac{1}{4}\); that there are two values between which it cannot lie when \(k\) is negative and numerically less than \(\frac{1}{4}\), or also when \(k>4\); and that there are two values between which it must lie in case \(k\) is positive and less than 4, these two values being coincident when \(k=1\).