Problems

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 \[ 2\sinh n\theta \{\omega\sinh n\theta - \sinh(n+2)\theta\}\{1-2\omega\cosh 2\theta + \omega^2\}^{-1}. \]

Two circles meet in the points \(A\) and \(B\) and tangents are drawn to them from a point \(P\) in their plane. Shew that if the four points of contact of the tangents are concyclic the point \(P\) must lie on the line \(AB\). Invert this theorem with respect to a circle whose centre is \(A\).

Shew that the two tangents to a conic which pass through a point are equally inclined to the lines which join the point to the foci of the conic. The points \(B\) and \(D\) are symmetrically placed with respect to \(AC\). Shew that the locus of the foci of conics which touch \(AB, BC, CD\) and \(DA\) is a circle and the line \(AC\).

Given three points \(A, B\) and \(C\) and two lines \(\alpha\) and \(\beta\) shew, by reciprocation and projection or otherwise, that there are in general four conics which pass through \(A, B\) and \(C\) and touch \(\alpha\) and \(\beta\). Shew further that the four chords of contact of these conics with \(\alpha\) and \(\beta\) form a quadrilateral whose diagonal triangle is \(ABC\).

Shew that if two coplanar triangles are in perspective from a point, called the centre of perspective, then the points of intersection of corresponding sides lie on a line, the axis of perspective. Three coplanar triangles are two by two in perspective. Shew that if they have the same centre of perspective the three axes of perspective are concurrent, and that if they have the same axis of perspective the three centres of perspective are collinear.