For \(m=-2\) and \(m=-\frac{1}{2}\) show that the curve \(r^m = a^m \cos m\theta\) becomes a rectangular hyperbola and a parabola respectively. In the curve \(r^m = a^m \cos m\theta\) show that \(r^{m+1}=a^m p\), where \(p\) is the perpendicular on the tangent at \((r, \theta)\).
\(P\) is a point near the origin on the curve \(y=x^2\). If \(\rho\) is the radius of curvature at \(P\) and if \(R\) is the radius of the circle through \(P\) touching the curve \(y=x^2\) at the origin, show that as \(P\) approaches the origin \(\text{Lt}\left(\frac{\rho}{R}\right) = \frac{1}{2}\).
A circle passes through a fixed point and determines an involution on a fixed straight line. Prove that its centre moves on a straight line.
From a point on the radius \(OA\) of the circumcircle of a triangle \(ABC\) perpendiculars are drawn to meet the sides in \(L, M, N\), and \(AD\) is perpendicular to \(BC\). Prove that \(L, M, N, D\) are concyclic points.
Prove that, if the equation \(\sqrt{(ax+b)} + \sqrt{(cx+d)}=e\) has equal roots, they are given by \((ax+b)/(cx+d)=1\) or \(a^2/c^2\).
Eliminate \(\theta\) from the equations \[ \frac{x}{\cos\theta+e\cos\alpha} = \frac{a}{\sin\theta}, \quad \frac{y}{1+e\cos(\theta+\alpha)} = b, \] where \(b^2=a^2(1-e^2)\).
The foot of a flagstaff is 19 feet above the eye-level. Its lower portion, 17 feet high, and its upper portion 18 feet high subtend equal angles at the eye. Prove that its horizontal distance from the eye is 144 feet.
Prove that in the ellipse the product of the focal perpendiculars on the tangent is constant. An ellipse of excentricity \(e\) has a focus at the origin, and touches the line \(x+c=0\). Prove that the other focus lies on a circle of radius \(2ce/(1-e^2)\).
Prove that the circle which has with the parabola \(y^2-4ax=0\) the common chords \(x+4y-5a=0, x-4y+7a=0\), passes also through the poles of these chords with respect to the parabola.
Prove that, in the rectangular hyperbola \(2xy=c^2\), the normals at the extremities of the chords \(x+2y-c=0, 4x-2y+3c=0\) meet in a point, and find the point.