Problems

The points \(P_1, P_2, \dots, P_n\) are the vertices of a regular \(n\)-agon inscribed in a circle \(C_0\), and \(A\) is a point on a concentric circle \(C_1\). Shew that \[ AP_1^2 + AP_2^2 + \dots + AP_n^2 \] remains fixed when \(A\) moves round the circle \(C_1\).

Shew that if \(u=(1-x^2)^n\), \[ u'(1-x^2) + 2nxu = 0; \] and by differentiating this equation \(n+1\) times, shew that \[ (1-x^2)\frac{d^2P}{dx^2} - 2x\frac{dP}{dx} + n(n+1)P=0, \] where \[ P(x) = \left(\frac{d}{dx}\right)^n \{(1-x^2)^n\}. \] By proceeding similarly with \[ v=e^{-x^2}, \] shew that if \[ H(x)e^{-x^2} = \left(\frac{d}{dx}\right)^n e^{-x^2}, \] then \[ \frac{d^2H}{dx^2} - 2x\frac{dH}{dx} + 2nH = 0. \]

Find the equations of the tangent and normal to the curve \(\phi(x,y)=0\) at the point \((x_0, y_0)\) on it. \par Shew that the normal distance of a point \((x_0, y_0)\) on \(\phi(x,y)=a\) from the curve \(\phi(x,y)=a+\delta\) is \[ |\delta|\left[\left(\frac{\partial\phi}{\partial x_0}\right)^2 + \left(\frac{\partial\phi}{\partial y_0}\right)^2\right]^{-\frac{1}{2}}, \] if \(\delta^2\) is neglected, and provided that \(\frac{\partial\phi}{\partial x_0}\) and \(\frac{\partial\phi}{\partial y_0}\) do not vanish.

Give an account of the method of finding the asymptotes of the curve \(P(x,y)=0\), where \(P\) is a polynomial in \(x\) and \(y\). \par Shew that \(x-y=3\) is an asymptote of \[ (x-y+1)(x-y-2)(x+y) = 8x-1, \] find the other asymptotes, and sketch the curve.

Shew that in the range \(a < x < b\), \[ \frac{d}{dx}\left( -2\tan^{-1}\sqrt{\frac{b-x}{x-a}} \right) = \frac{1}{\sqrt{(b-x)(x-a)}}, \] and integrate with respect to \(t\) \[ \frac{1}{(1-kt)\sqrt{1-t^2}} \quad (0

Shew that if \[ I_m = \int_0^\infty e^{-x}\sin^m x dx \] and \(m\ge 2\), then \[ (1+m^2)I_m = m(m-1)I_{m-2}; \] and hence evaluate \(I_4\).

The straight lines \(AB\) and \(CD\) intersect in \(U\). \(AC\) and \(BD\) in \(V\); \(UV\) intersects \(AD\) and \(BC\) in \(F\) and \(G\) respectively; \(BF\) intersects \(AC\) in \(L\). Prove that \(LG, CF\) and \(AU\) meet in a point.

Prove that the polar reciprocal of a conic with regard to a focus is a circle. \par Find the number of conics which have a given focus and pass through three given points.

The tangents at points \(P\) and \(Q\) of a parabola meet at \(T\), and are of equal length. From a point \(U\) on \(TP\) another tangent is drawn, cutting \(TQ\) in \(V\). Prove that \(TV=PU\).

Any two conjugate diameters of an ellipse meet the tangent at one end of the major axis in \(Q\) and \(R\). Prove that \(QR\) subtends supplementary angles at the foci.