Eliminate \(x\) and \(y\) from the equations \[ ax^2+by^2=1, \quad a'x^2+b'y^2=1, \quad lx+my=1, \] obtaining the result in a rational form.
Shew that the sum of the homogeneous products of \(a,b,c\), of \(n\) dimensions is \(\Sigma a^{n+2}/(b-a)(c-a)\).
\(ABCD\) is a horizontal line and \(DE\) a vertical line. \(DE\) subtends angles \(\theta, 2\theta, 3\theta\), at \(A, B, C\), respectively. Find \(CD\) and \(DE\) when \(AB=260, BC=100\).
Find the sum of the series \[ 1 + x\cos\theta + x^2\cos 2\theta + \dots \text{ to } \infty, \quad (|x|<1). \] If \(\alpha, \beta\) are angles less than \(\pi\) and such that \(\tan\frac{\alpha}{2} = 2\tan\frac{\beta}{2}\), prove that \[ \frac{\alpha-\beta}{2} = \frac{1}{1 \cdot 3}\sin\beta + \frac{1}{2 \cdot 3^2}\sin 2\beta + \frac{1}{3 \cdot 3^3}\sin 3\beta + \dots. \]
Find the condition of perpendicularity of two straight lines whose equations are given in trilinear co-ordinates.
Find the equations of the tangent and normal at any point of the curve \[ x = a\cos^3\alpha, \quad y=a\sin^3\alpha. \] If the normal at the point \(\alpha\) is the tangent at the point \(\beta\), prove that \(\tan\alpha\) and \(\tan\beta\) have each one of the values \((\pm\sqrt{5}\pm 1)/2\).
Find the asymptotes of the curve \[ x(x^2-y^2)+x^2+y^2+x+y=0. \] Shew that the asymptotes meet the curve again in three points on a straight line, and find the equation of the line.
Prove the formulae \(\rho = r \frac{dr}{dp}\) and \(\frac{1}{p^2} = \frac{1}{r^2} + \frac{1}{r^4}\left(\frac{dr}{d\theta}\right)^2\). Find the radius of curvature at any point of the cardioid \(r=a(1+\cos\theta)\).
Find formulae of reduction for \[ (1) \int \sin^m\theta \cos^n\theta d\theta, \quad (2) \int x^n(a+bx)^p dx. \]
Find the whole area of \(a^4y^2=x^5(2a-x)\) and the area of a loop of \(x^4+y^4=2a^2xy\).