If R and r are the radii of the circumscribed circle and inscribed circle of a triangle ABC, prove that \[ r = 4R\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}, \] and that the distance between their centres is \((R^2-2Rr)^{\frac{1}{2}}\). Prove also that the product of the distances of the centre of the inscribed circle from the centres of the three escribed circles is \(16R^2r\).
Sum the series
Express \(x^{2n}-2x^n\cos n\theta+1\) as the product of \(n\) real quadratic factors, and deduce that \[ \sin n\theta = 2^{n-1}\sin\theta\sin(\theta+\frac{\pi}{n})\sin(\theta+\frac{2\pi}{n})\dots\sin(\theta+\frac{n-1}{n}\pi). \] Prove that \[ \cot\theta+\cot(\theta+\frac{\pi}{n})+\cot(\theta+\frac{2\pi}{n})+\dots+\cot(\theta+\frac{n-1}{n}\pi) = n\cot n\theta. \]
Prove that \[ x^2+y^2-hx-ky = \lambda\left(\frac{x}{h}+\frac{y}{k}-1\right) \] is the general equation of a circle passing through the two points \((h,0)\) and \((0,k)\); and that the circles corresponding to the values \(\lambda_1\) and \(\lambda_2\) of \(\lambda\) cut one another at right angles if \(\lambda_1\lambda_2 + h^2k^2=0\).
Find the equation of the polar of the point \((h,k)\) with regard to the parabola \(y^2=4ax\). Circles are drawn touching a parabola at a given point, and cutting it at two other points, prove that the locus of the pole of the chord joining these points is a straight line.
Shew that, if \(\frac{x}{a} = \frac{y-b\lambda}{\gamma\lambda} = \frac{b-\gamma\lambda}{b}\), the locus of the point \((x,y)\) is the ellipse \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1. \] Shew also that \(x(1-\lambda_1\lambda_2)+\frac{y}{b}(\lambda_1+\lambda_2)=1+\lambda_1\lambda_2\) is a straight line joining two points on the ellipse, at which \(\lambda\) has the values \(\lambda_1\) and \(\lambda_2\); and that the tangents at those points meet at the point \((\frac{1-\lambda_1\lambda_2}{1+\lambda_1\lambda_2}a, \frac{\lambda_1+\lambda_2}{1+\lambda_1\lambda_2}b)\).
Define conjugate lines with respect to a conic. Find the condition that \(lx+my+nz=0\) and \(l'x+m'y+n'z=0\) shall be conjugate with respect to the conic \(uyz+vzx+wxy=0\). Hence or otherwise, find the equation of the locus of a point P, such that AP and BP are conjugate lines with respect to the given conic, where ABC is the triangle of reference.
Differentiate \(\tan^{-1}\frac{4x^{\frac{1}{2}}}{1-x}\) with respect to \(x\). If \(x=y\log xy\), find \(\frac{dy}{dx}\). Prove that the \(n\)th differential coefficient of \(\tanh^{-1}\frac{x}{a}\) is \[ \frac{(n-1)!}{2}\left[\frac{(-1)^{n-1}}{(a+x)^n}+\frac{1}{(a-x)^n}\right]. \]
Explain the meaning of partial differentiation. If \(f(x,y)=0\) and \(\phi(y,z)=0\), shew that \[ \frac{\partial f}{\partial y}\frac{\partial\phi}{\partial z}\frac{dz}{dx} = \frac{\partial f}{\partial x}\frac{\partial\phi}{\partial y}. \]
Obtain the usual differential equation \(EI\frac{d^4y}{dx^4}=w\) for the deflection of a uniform heavy beam when the deflection is everywhere small. A uniform beam of length \(l\) is supported at the ends; prove that the greatest droop occurs at the middle point and is \[ \frac{5wl^4}{384EI}. \]