Prove that a circle can be projected orthogonally into an ellipse, and give examples of properties of an ellipse derived by this method from corresponding properties of a circle.
A, B, C, D are the vertices of a tetrahedron in which the straight line joining \(A\) to the orthocentre of the triangle \(BCD\) is perpendicular to the plane \(BCD\). Prove that \[ AB^2+CD^2=AC^2+BD^2=AD^2+BC^2. \] Determine whether the converse of this theorem is also true.
Obtain the condition that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] shall represent some pair of straight lines \(l_1, l_2\). If also the equation \(a_1x^2+2h_1xy+b_1y^2+2g_1x+2f_1y+c_1=0\) represents a pair of straight lines \(l_2, l_3\) (so that \(l_2\) is a line common to the two pairs), shew that \begin{align*} (a_1c-ac_1)^2 &= 4(g_1c-gc_1)(a_1g-ag_1), \\ \text{and} \quad (b_1c-bc_1)^2 &= 4(f_1c-fc_1)(b_1f-bf_1). \end{align*} Shew further that the coordinates of the point of intersection of \(l_1\) and \(l_3\) are \[ \left( \frac{(bc_1-cb_1)(ca_1-ac_1)}{2(ab_1-ba_1)(cg_1-gc_1)}, \frac{(bc_1-cb_1)(ca_1-ac_1)}{2(ab_1-ba_1)(fc_1-cf_1)} \right). \]
If \(u=0, v=0\) are the equations of two straight lines, find the equation of the harmonic conjugate of \(u+\lambda v=0\) with respect to the two given lines, \(\lambda\) being a constant. \(T\) is a given point on a circle whose centre is \(O\) and whose radius is \(a\). From a point \(P\) on the circle a straight line is drawn to intersect, at a fixed acute angle \(\tan^{-1}m\), the tangent at \(T\) to the circle, the point of intersection being \(Q\). Prove that for all positions of \(P\) on the circle, the harmonic conjugate of \(QP\) with respect to \(QT, QO\), passes through a fixed point \(A\) at a distance \(\frac{a}{2m}\sqrt{1+m^2}\) from \(O\). Shew also that the angle between the extreme positions of the line \(AQ\) is \[ \pi - \cos^{-1}\sqrt{\frac{9+9m^2}{9+25m^2}}. \]
Find the equation of the ellipse which passes through the origin, which has the point \((0,4)\) as one focus, and such that its minor axis lies along the line whose equation is \(x+2y=3\). Shew that the equations of the equiconjugate diameters of this ellipse are \[ 8x+y+6=0 \quad \text{and} \quad 4x-7y+18=0 \] respectively.
Obtain the equation of the rectangular hyperbola which touches the conic \[ ax^2+by^2+1=0 \] at each of the points in which it is cut by the line \(lx+my+n=0\). If \(l,m,n\) vary in such a way that the rectangular hyperbolas corresponding to the different positions of the straight line always pass through the origin, prove that the line \(lx+my+n=0\) touches a fixed circle whose centre is the origin. Prove also that in this case the envelope of the polar with respect to the corresponding rectangular hyperbola of the point of contact of \(lx+my+n=0\) with this circle is the ellipse \[ a^2x^2+b^2y^2 = a+b. \]
The polar equation of a conic is written in the form \(\frac{l}{r}=1+e\cos\theta\). Interpret the constants \(l,e\) and find the equation of the tangent to the conic at the point \(\theta=\alpha\). \(S\) is one focus and \(V\) the nearer extremity of the major axis of an ellipse whose latus rectum is of length \(kl\). \(S\) is also the vertex of a parabola whose axis lies along \(VS\) produced and the length of whose latus rectum is \(l\). If \(P\) is one point of intersection of the ellipse and parabola shew that the cosine of the angle \(VSP\) satisfies the equation in \(x\) \[ x^2(2e-k)+2x+k=0, \] and that in the case \(2e=4-3k\), (\(2<3k<3\)), the tangent at \(P\) to the ellipse cuts \(SV\) produced at a distance \(\frac{kl}{3(1-k)}\) from \(S\). How is this result affected if \(3<3k<4\)?
If the median from the vertex \(B\) of an acute-angled triangle \(ABC\) makes an angle \(\alpha\) with \(BA\), prove that \(\cot\alpha=2\cot B+\cot A\). Shew also that the area of the triangle formed by the bisector of the angle \(A\), the median from \(B\), and the perpendicular from \(C\) on to \(AB\) is \[ \frac{b^2\tan\alpha\tan^2\frac{A}{2}(\tan^2\frac{A}{2}\sin A\cot\alpha - \cos A)^2}{8(1+\tan^2\frac{A}{2}\cot\alpha)^2}. \]
Establish the result \((\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta\) for the case when \(n\) is an integer, and state the corresponding result for the hyperbolic functions \(\cosh u, \sinh u\). Shew that the \(n\)th roots of unity can be written in the form \(1, \omega, \omega^2, \dots, \omega^{n-1}\) and that the sum of the series \[ \cosh 2\theta + \omega\cosh 4\theta + \omega^2\cosh 6\theta + \dots + \omega^{n-1}\cosh 2n\theta \] is \[ 2\sinh n\theta \{\omega\sinh n\theta - \sinh(n+2)\theta\}\{1-2\omega\cosh 2\theta + \omega^2\}^{-1}. \]