\(OC\) touches a circle at \(C\) and \(OAB\) is a chord. Prove that \[ AB:OC :: BC^2-AC^2 : BC.AC. \] Prove also that, if \(OC\) is equal to the radius, the triangle \(ABC\) is greatest when the angle \(COA\) is \(60^\circ\).
The lower part of a flagstaff, of height \(a\), and the upper part, of height \(b\), subtend equal angles \(\theta\) at the top of a pedestal, of height \(c\), and at distance \(x\). Prove that \[ (b-a)x^2=(a-c)\{a^2+a(b-c)+bc\}, \text{ and that } \frac{2ab\sin^2\theta}{b-a} = a+\frac{c(a+b)}{a+b-2c}. \]
Justify the formula for measuring the length of an arc of a circle. `From \(\frac{8}{3}\) of the chord of half the arc subtract \(\frac{1}{3}\) of the chord of the whole arc.' Estimate the error involved, and, by applying the formula to an arc of \(60^\circ\), find \(\sin 15^\circ\) correct to 4 decimal places.
Trace roughly the curves \(x^2-y=2\) and \((y-3)(x+1)+8=0\) between \(x=-4\) and \(x=4\). Use your figure to find their points of intersection.
Find the limit as \(x \to a\) of \((x^n-a^n)/(x-a)\) for commensurable values of \(n\), whether positive or negative, and apply the result to the differentiation of \(x^n\). Prove that \[ \frac{d^4}{dx^4}(x^a e^x) = a^x e^x. \]
Use Maclaurin's Theorem to expand \(e^{-\cos x}\) in ascending powers of \(x\).
Express in partial fractions, and integrate with respect to \(x\), the expression \[ \frac{x^4+4x^2+9x}{(x+1)^2(x^2+1)}. \]
Evaluate \(\int_0^\infty \frac{dx}{\sqrt{x(4-x)(x-3)}}\) and \(\int_0^\infty \frac{dx}{(2+x)\sqrt{x(1+x)}}\). Shew without integration that \(\int_0^{\frac{2\pi}{3}} \frac{64d\theta}{(5+3\cos\theta)^2}\) lies between 644 and 753; and, by integrating, that its value is about 68. (Take arc tan \(\frac{1}{2}=.322\) and arc tan \(\frac{1}{4}=.165\).)
Prove that the feet of the perpendiculars let fall from a point on the circumcircle of a triangle on the three sides lie on a straight line (the pedal line of the point). Prove that, if three points on the circumcircle are at the vertices of an equilateral triangle, their pedal lines form an equilateral triangle.
Prove that the inverse of a circle is either a straight line or a circle. Two circles whose centres are \(O, O'\) cut orthogonally. \(Q, Q'\) are the points inverse to a point \(P\) with regard to the two circles respectively. \(OQ'\) and \(O'Q\) meet in \(R\). Prove that \(P, Q, R, Q'\) lie on a circle.