Prove that the inverse of a circle with respect to a point in its plane is a circle or a straight line. Two variable circles touch each of two fixed circles and touch each other; shew that the locus of the point of contact of the variable circles is two orthogonal circles coaxal with the fixed ones.
Prove that the polar reciprocal of a circle with respect to any circle is a conic. Reciprocate a system of coaxal circles with respect to a circle whose centre is one of the limiting points. Prove, by reciprocation, that an ellipse and a confocal hyperbola cut at right angles; and that the tangent at any point \(P\) of the hyperbola makes equal angles with the tangents from \(P\) to the ellipse.
Shew that if the sum of the squares on a pair of opposite edges of a tetrahedron is equal to the sum of the squares on a second pair of opposite edges, then the two remaining edges are perpendicular to each other.
Prove that the equation \(S=x^2+y^2+2gx+2fy+C=0\) represents a circle, and examine the meaning of \(S\) when \((x,y)\) lies (1) without, (2) within the circle. Find the equation of the circle whose diameter is the chord of intersection of \(S=0\) with \(lx+my+n=0\).
Find the equation of the normal at any point of the parabola \(y^2=4ax\). A triangle is formed by the normal at any point of the parabola and by the two tangents at the points where it cuts the parabola. Prove that the locus of its centroid is \[ y^4 - 3axy^2 + 2a^2y^2 + 4a^4=0. \]
Find the conditions that the normals to \(x^2/a^2+y^2/b^2=1\) at its points of intersection with \(lx+my-1=0\) and \(l'x+m'y-1=0\) should be concurrent. From any point on this ellipse three normals are drawn and their feet are \(P, Q, R\). Prove that if \(P\) is \((a\cos\phi, b\sin\phi)\), then \(QR\) is \[ x\cos\phi/a^3 - y\sin\phi/b^3+1/(a^2-b^2)=0. \]
Find the condition that \(lx+my+n=0\) should touch the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] A chord of a conic subtends a right angle at a fixed point: prove that (i) the foot of the perpendicular from the point on the chord is a circle, and (ii) the chord envelopes a conic and find in what case this envelope is a parabola.
Find, in trilinear coordinates, the locus of the centres of conics touching the sides of the triangle of reference and the line \(l\alpha+m\beta+n\gamma=0\).
Solve the equations:
Expand \(\log_e(1+x)\) in powers of \(x\), when \(|x|<1\). Verify that \(6^9\) is roughly equal to a power of 10, and, taking \(\log_{10} e = \cdot 434\), prove that \(\log_{10}6 = \cdot 77815\).