Problems

Prove that there are five, and only five, types of regular polyhedrons. Calculate the number of edges, faces and vertices that each type has.

Prove that there is a unique parabola touching four given fixed lines in general position in a plane. \newline Explain and justify geometrical constructions for its focus and vertex, and for the points of contact with the given lines.

State, without proof, the relationship between the positions of the circumcentre, centroid and orthocentre of a triangle. \newline Prove that if a rectangular hyperbola circumscribes a triangle it passes through the orthocentre. \newline A circle of variable radius with fixed centre \((3a, 3b)\) cuts the rectangular hyperbola \(xy=9c^2\) in the points \(A, B, C, D\). Prove that the locus of the centroid of the triangle \(ABC\) is the rectangular hyperbola \((x-2a)(y-2b)=c^2\).

If \(A, B, C, D\) are four fixed points on a conic and \(P\) a variable point on the conic, prove that the cross-ratio of the pencil \(P(ABCD)\) is independent of the position of \(P\). \newline State the dual theorem. \newline A conic is inscribed in a triangle \(ABC\), and \(D\) is the point of contact with \(BC\). The tangent parallel to \(BC\) touches the conic at \(E\), and \(AE\) meets \(BC\) in \(F\). Prove that the mid-point of \(DF\) is the mid-point of \(BC\).

Prove that the conic \[ x^2 - 4xy + 4y^2 + 12x - 4y + 6 = 0 \] is a parabola. Find the coordinates of its focus and the equations of its directrix and tangent at the vertex.

If the polar equation of a conic is \(l/r = 1+e\cos\theta\), show that the equation of the chord joining the two points \(\theta=\alpha-\beta, \theta=\alpha+\beta\), is \[ \frac{l}{r} = e\cos\theta + \sec\beta\cos(\theta-\alpha). \] Find also the equation of the tangent at a given point. \newline Prove that chords of a conic subtending a constant angle at a focus \(S\) envelop a conic, and that the poles of these chords with respect to the given conic lie on another conic. Prove also that these two derived conics have \(S\) as focus and the corresponding directrix of the original conic as directrix.

Prove that \[ \tan \frac{\pi}{5} = \sqrt{5} \tan \frac{\pi}{10}, \] and hence, or otherwise, show that \[ \tan \frac{\pi}{20} = \frac{1}{4}(\sqrt{5}+1)\{4-\sqrt{(10+2\sqrt{5})}\}. \]

\(X\) is the point inside a triangle \(ABC\) such that \(XB, XC\) are the internal trisectors of the angles \(B, C\) adjacent to the side \(BC\), and \(Y, Z\) are similarly defined in relation to \(CA, AB\) respectively. Prove that \[ AY:AZ = \sin(60^\circ+\frac{1}{3}B) : \sin(60^\circ+\frac{1}{3}C). \] Hence, or otherwise, show that the triangle \(XYZ\) is equilateral, and express the length of its sides in symmetrical form.

(i) Prove that \[ \frac{d^3x}{dy^3} = -\frac{d^3y}{dx^3} \left(\frac{dx}{dy}\right)^4 + 3\left(\frac{d^2y}{dx^2}\right)^2 \left(\frac{dx}{dy}\right)^5. \] % Note: The OCR for this question was garbled. The above is a standard identity that fits the visible structure. (ii) If \(ax^2+2hxy+by^2+2gx+2fy+c=0\), show that \[ \frac{d^2y}{dx^2} = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} (hx+by+f)^{-3}. \]

Find an expression for \(\frac{d^n}{dx^n} \tan^{-1}x\). \newline Prove that when \(x=0\) its value is zero if \(n\) is even and \(\pm(n-1)!\) if \(n\) is of the form \(4p\pm 1\), where \(p\) is an integer. \newline Hence obtain the series expansion of \(\tan^{-1}x\).