Problems

A particle moves in a plane so that its position at time \(t\), referred to fixed rectangular cartesian axes, is given by \(x = a \sin 2pt\), \(y = a \sin pt\). Sketch the path traced out by the particle, and find the radii of curvature at the points where the particle is moving in a direction parallel to one or other of the axes.

1. Find the indefinite integrals \[ \int \frac{(1+x^2) \, dx}{x^2(1-x)}, \quad \int \cos^4 x \, dx. \]
2. Evaluate the integral \[ \int_a^b x^2\sqrt{\{(x-a)(b-x)\}} \, dx, \] where \(0

\(P\) is a variable point \((at^2, 2at)\) and \(K\) is the fixed point \((ak^2, 2ak)\) of the parabola \(y^2=4ax\). The foot of the perpendicular from \(P\) to \(OK\) is \(M\), where \(O\) is the origin. Prove that, if \(OM=r\), \[ \frac{dr}{dt} = \frac{2(kt+2)a}{\sqrt{(4+k^2)}}. \] Prove also that, if the arc \(OK\) is rotated about \(OK\), the volume of the solid generated is \(2\pi k^5 a^3/15\sqrt{(4+k^2)}\).

Prove that, if \(P, A, B, C\) are four points in a plane, there is another point \(P'\) in the plane such that \[ P'A:P'B:P'C = PA:PB:PC, \] and that \(P'\) coincides with \(P\), if and only if \(P\) lies on the circle \(ABC\).

Prove that the inverse of a straight line is a circle through the centre of inversion. Circles \(BDC, CEA, AFB\) cut the circle \(ABC\) orthogonally. Through \(A, B, C\) circles \(AD, BE, CF\) are drawn orthogonal to \(ABC\) and to \(BDC, CEA, AFB\) respectively. Prove that the circles \(AD, BE, CF\) meet in two points.

Prove that the polar reciprocal of one circle with regard to another is a conic. A conic is drawn with a focus at the centre \(C\) of a given circle so as to touch the circle and pass through a fixed point \(O\) on the circumference. Prove that the corresponding directrix touches the circle on \(CO\) as diameter.

\(A, B, C, D\) are the common points of two conics \(S, S'\). Prove, by projection or otherwise, that if the tangents to \(S\) at \(A, B\) meet on \(S'\), then the tangents to \(S\) at \(C, D\) also meet on \(S'\). State the dual theorem.

Prove that a chord of an ellipse which subtends a right angle at a given point \(P\) on the curve cuts the normal at \(P\) in a fixed point \(Q\). Prove that if the equation of the ellipse be \(x^2/a^2+y^2/b^2=1\), the locus of \(Q\) corresponding to different positions of \(P\) is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{(a^2-b^2)^2}{(a^2+b^2)^2}. \]

Find the conditions that the lines \(lx+my+n=0\), \(l'x+m'y+n'=0\) may be conjugate diameters of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \quad (ab \ne h^2). \] Prove that the equation of the pair of lines which are conjugate diameters of both the conics \(ax^2+2hxy+by^2=1\) and \(a'x^2+2h'xy+b'y^2=1\) is \[ (ah'-a'h)x^2+(ab'-a'b)xy+(hb'-h'b)y^2=0. \]

Prove that, if \(A\) is any point on a conic and \(PQR\) is a self-conjugate triangle and \(AQ, AR\) meet the conic in \(B, C\), then \(P, B, C\) are collinear. Prove that, if \(ABC\) is a triangle inscribed in a conic, then there is an infinite number of self-conjugate triangles \(PQR\) such that \(P\) lies on \(BC\), \(Q\) on \(CA\) and \(R\) on \(AB\). Prove that \(AP, BQ, CR\) intersect on the conic.