The roots of the quadratic equation \(az^2+2bz+c=0\), where \(a, b, c\) are real and \(ac>b^2\), are represented on an Argand diagram by points \(P, Q\). Prove that \(P\) and \(Q\) are equidistant from the origin, and that \(PQ\) is perpendicular to the axis of real numbers. Hence show that \(P\) and \(Q\) may be found by a geometrical construction which does not require the solution of the equation. Prove also that, if \(a', b', c'\) are real and \(a'c'>b'^2\), the points representing the roots of \(a'z^2+2b'z+c'=0\) lie on the circle through \(P, Q\) and the origin, if \(bc'=b'c\).
A rod \(AB\) moves so that \(A, B\) respectively lie on fixed lines \(OP, OQ\) inclined at an angle \(\alpha\). Prove that, if \(OA=x, OB=y\), and the area of \(OAB=u\), \[ \frac{du}{dx} = \frac{(y^2-x^2)\sin\alpha}{2(y-x\cos\alpha)}. \] Deduce that \(u\) is a maximum when \(x=y\).
Find the equation of the tangent at any point of the curve \(x=f(t), y=F(t)\). If \(Y\) is the foot of the perpendicular from the origin \(O\) on the tangent to the curve \(ay^2=x^3\) at any point \(P(at^2, at^3)\), show that the coordinates of \(Y\) are \(\dfrac{3at^4}{9t^2+4}\) and \(\dfrac{2at^3}{9t^2+1}\). Deduce that, if \(OY=p\) and the inclination of \(OY\) to the axis of \(x\) is \(\psi\), \[ 27p = 4a\cos\psi\cot^2\psi. \]
Explain what is meant by a point of inflexion on a plane curve, and prove that, if \(y=f(x)\) has a point of inflexion whose abscissa is \(x_0\), \(f''(x_0)=0\). The graph of a polynomial of the fourth degree in \(x\) touches the \(x\)-axis at \((a,0)\), and has a point of inflexion at \((-a,0)\). Prove that the graph passes through \((-2a,0)\), and that it has a second point of inflexion whose abscissa is \(a/2\).
\(AB\) is a diameter of a circle whose centre is \(O\); \(ODC\) and \(BEC\) are straight lines cutting the circle in \(D\) and \(E\) respectively; \(OB\) is produced to \(F\) so that \(OF=OC\); \(AE\) produced intersects \(CF\) in \(G\). Prove that \(DG\) is a tangent to the circle.
Shew how to determine by a geometrical construction the focus and directrix of a parabola of which two parallel chords are completely given.
Three conics \(A, B, C\) touch two given straight lines. \(P\) is the intersection of the other common tangents of \(B\) and \(C\), \(Q\) that of the other common tangents of \(C\) and \(A\), \(R\) that of the other common tangents of \(A\) and \(B\). Prove that \(P, Q\) and \(R\) are collinear.
Shew that, if the equation of a circle in areal coordinates is in the form \[ \phi(x,y,z) \equiv a^2yz + b^2zx + c^2xy - (lx+my+nz)(x+y+z) = 0, \] the square of the length of a tangent from an external point \((x,y,z)\) is \(-\phi(x,y,z)\). Shew also that this circle will cut the circumscribing circle orthogonally if \[ la\cos A + mb\cos B + nc\cos C = abc. \]
If \begin{align*} a(y^2+z^2-x^2) &= b(z^2+x^2-y^2) = c(x^2+y^2-z^2), \\ \text{and } x(b^2+c^2-a^2) &= y(c^2+a^2-b^2), \end{align*} prove that \[ a^3+b^3+c^3 = (b+c)(c+a)(a+b). \]
If \[ x = \cfrac{1}{a_1 + \cfrac{1}{a_2 + \dots + \cfrac{1}{a_{r-1} + \cfrac{1}{a_r + \cfrac{1}{a_{r+1} + \dots + \cfrac{1}{a_n}}}}}}, \] prove that the continued fraction \[ \cfrac{1}{a_1 + \dots + \cfrac{1}{a_{r-1} + \cfrac{1}{a_{r+1} + \dots + \cfrac{1}{a_n}}}}, \] where the constituent \(a_r\) is omitted, is with the usual notation equal to \[ \frac{x(p_{r-1}q_{r-1} - p_{r-2}q_r) + p_{r-2}p_r - p_{r-1}^2}{x(q_{r-1}^2 - q_{r-2}q_r) + p_r q_{r-2} - p_{r-1}q_{r-1}}. \]