Problems

A chord \(PQ\) of a parabola passes through the focus. Prove that the circle on \(PQ\) as diameter touches the directrix.

A conic is drawn touching an ellipse at ends \(A, B\) of its axes, and passing through the centre \(C\) of the ellipse; prove that the tangent at \(C\) is parallel to \(AB\).

Through two given points \(A, B\) a variable circle is drawn, and either arc \(AB\) is trisected at \(P\) and \(Q\). Prove that the loci of \(P\) and \(Q\) are branches of two hyperbolas, each of which passes through the centre of the other.

Find the equation of the circle circumscribing the triangle formed by the lines \(ax^2 + 2hxy+by^2=0\) and \(y=k\), and prove that the tangent at the origin is \[ 2hx = (a-b)y. \]

If two normals to the parabola \(y^2=4ax\) make complementary angles with the axis, prove that their point of intersection lies on one of the parabolas \[ y^2 - ax + 2a^2 = \pm a^2. \]

Prove that the locus of the poles of normal chords of the ellipse \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) is the curve \[ \frac{a^6}{x^2} + \frac{b^6}{y^2} = (a^2-b^2)^2. \]

Two adjacent corners \(A, B\) of a rigid rectangular lamina \(ABCD\) slide on the rectangular axes \(XOX'\), \(YOY'\), all the motion being in one plane. Prove that the locus of \(C\) is an ellipse of area \(\pi AD^2\).

Prove that the three lines, each of which forms a harmonic pencil with the three lines \[ y=0, \quad ax^2+2hxy+by^2=0, \] are \[ ax+hy=0, \quad a(ax^2+2hxy+by^2)+8(ab-h^2)y^2=0. \]

If the equations \begin{align*} axy+bx+cy+d&=0, \\ ayz+by+cz+d&=0, \\ azw+bz+cw+d&=0, \\ awx+bw+cx+d&=0, \end{align*} are satisfied by values of \(x, y, z, w\) which are all different, show that \[ b^2+c^2=2ad. \]

Prove that the number of primes is infinite. Find \(n\) consecutive numbers, none of which are primes.