Find the equation of the tangent at any point of a given curve. \par Prove that in the lemniscate, \((x^2+y^2)^2=2a^2(x^2-y^2)\), the tangent at a point for which \(y=\frac{2}{3}x\) is \(117y-44x=7a\sqrt{14}\).
Integrate the following expressions with respect to \(x\) \[ \frac{1}{\sqrt{(x^2-a^2)}}, \quad \frac{1}{2\sqrt{\{(2-x)(x-1)\}}}, \quad \frac{20}{25+9\cos x+12\sin x}. \]
Given two circles and a point \(A\) on one of them, shew how to draw a chord \(BA\) of one circle such that if produced to meet the other circle in \(C\), \(BA\) may be equal to \(AC\).
Prove that the inverse of a circle is in general another circle. \par If \(P, Q\) are inverse points with regard to a circle \(A\) and the figure is inverted with regard to any centre, \(P', Q'\) and \(A'\) being the corresponding elements in the inverse figure, then \(P', Q'\) are inverse points with regard to \(A'\).
If the normal at \(P\) to a hyperbola meet the axes in \(G\) and \(g\), prove that the ratio \(PG:Pg\) is constant. \par Prove that the normal and tangent to a hyperbola at any point meet the axes and asymptotes respectively in four points which lie on a circle through the centre of the conic.
Prove that the polar reciprocal of a circle with regard to another circle is a conic section. \par \(AB, AC, BC\) are three tangents to a conic the point \(A\) being on another conic having the same focus and directrix. Shew that the angle that \(BC\) subtends at the common focus is constant.
Explain the method of proving propositions by projection, stating what classes of properties are projective and illustrating by examples.
Prove that for different values of \(p\) the centroid of the triangle whose sides are \[ x\cos\alpha+y\sin\alpha-p=0 \] and \[ ax^2+2hxy+by^2=0 \] lies on the line \[ x(a\tan\alpha-h)+y(h\tan\alpha-b)=0. \]
Find the equation of the normal at any point on the curve \[ x=am^2, \quad y=2am. \] Shew that the normals to the curve at the extremities of the chords \[ y=4c, \quad ax+cy+\kappa^2=0 \] are concurrent.
Find equations for determining the foci of the conic represented by the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Find the foci of the curve \[ 3x^2+3y^2-2xy-8x+8y-24=0, \] and draw the curve.