Sketch the curve \(r(\cos\theta + \sin\theta) = a \sin 2\theta\), and find the area of the loop of the curve.
\(AA'\) is the transverse axis of a hyperbola, straight lines \(A'P, AP'\) are drawn through \(A', A\) parallel to the asymptotes, \(PAP'\) cuts the hyperbola again in \(Q\). Prove that \(Q\) is the middle point of \(PP'\).
If \begin{align*} X &= ax^2+2hxy+by^2, \\ Y &= a'x^2+2h'xy+b'y^2, \end{align*} and prove that \[ (h'^2-a'b')X^2 + (ab'+a'b-2hh')XY + (h^2-ab)Y^2 \] is the perfect square of a function of \(x\) and \(y\).
Prove that \[ \frac{a^2}{2a+} \frac{a^2}{b^2-2a+} \frac{a^2}{b^2-2a+} \dots \text{ to infinity} \] is the square of \[ \frac{a}{b+} \frac{a}{b+} \frac{a}{b+} \dots \text{ to infinity}. \]
Prove that \((1+\cos 11\theta)/(1+\cos\theta)\) is the square of a rational function of \(\cos\theta\), and find this function.
Eliminate \(\theta, \phi\) from the equations: \[ a\sec\theta+b\text{cosec }\theta=c, \quad a\sec\phi+b\text{cosec }\phi=c, \quad \theta+\phi=2\alpha. \]
Prove that, if the inscribed circle of a triangle subtends angles \(2\theta_1, 2\theta_2, 2\theta_3\) at the centres of the escribed circles, \[ 16R^2\sin\theta_1\sin\theta_2\sin\theta_3=r^2. \]
If the point \(P\) on \(x^2/a^2+y^2/b^2=1\) and the point \(P'\) on \(x^2/a'^2+y^2/b'^2=1\) have both the same eccentric angle, prove that the greatest possible angle between the normals at \(P\) and \(P'\) is \[ \tan^{-1}\{(ab'-a'b)/2(aa'bb')^{1/2}\}. \]
If \(2c\) is the distance between the foci of a system of confocal ellipses, prove that the locus of the centres of curvature at the points in which the system is met by the confocal hyperbola whose semi-transverse axis is \(c\cos\alpha\) is \[ \cos^6\alpha/x^2 - \sin^6\alpha/y^2 = 1/c^2. \]
If \(n\) is the length of the normal intercepted between a point \((r,\theta)\) of the curve \[ r^2=a^2\cos 2\theta \] and the initial line, and \(\rho\) is the radius of curvature at the point, prove that \[ n(2r+3\rho)=3r\rho. \]