Prove that the angles made by a tangent to a circle with a chord drawn from the point of contact are respectively equal to the angles in the alternate segments of the circle. Two circles touch at \(A\), one being inside the other; the tangent at a point \(B\) to the inner circle cuts the outer circle in \(C\) and \(D\). Prove that the angles \(BAC\) and \(BAD\) are equal.
Prove that, if the polar of a point \(P\) with respect to a circle passes through the point \(Q\), the polar of \(Q\) will pass through the point \(P\). Prove also that the circle described on \(PQ\) will cut the given circle at right angles.
Shew how to draw a perpendicular to a plane from a point outside it. Prove that if two straight lines neither intersect nor are parallel, there is one straight line perpendicular to both of them.
Prove that the two tangents drawn from a point to a parabola subtend equal angles at the focus. The normal at a point \(P\) in a parabola meets the curve again at \(Q\), and the tangents at \(P\) and \(Q\) intersect at \(T\). If \(S\) is the focus and \(K\) is the middle point of \(TQ\), prove that \(TSK\) is a right angle.
Prove that the latus-rectum of the conic, in which a given right-circular cone is cut by a plane, is proportional to the perpendicular to the plane from the vertex of the cone.
Shew that the equation \(axy+bx+cy+d=0\) may be written in the form \[ \frac{x-p}{x-q} = \lambda \frac{y-p}{y-q}, \] where \(\lambda\) has two values of which the product is \(-1\).
Find the general term in the expansion in powers of \(x\) of the expression \[ \frac{1-2x-x^2}{(1-x^2)(1+x+x^2)}. \]
Sum the series \(n^2+2(n-1)^2+3(n-2)^2+\dots\) where \(n\) is a positive integer; and find the \(n\)th term of the recurring series \(1+3x+10x^2+36x^3+\dots\).
Prove that any convergent to \(a_1+\frac{1}{a_2+}\frac{1}{a_3+}\dots\) is nearer to the continued fraction than any convergent preceding it. Shew that the \(n\)th convergent to the continued fraction \(1-\frac{1}{2-}\frac{1}{2-}\dots\) is equal to \(\frac{n}{n-1}\).
Five equal light rods are jointed together to form a regular pentagon \(ABCDE\) and two light rods \(BE\) and \(BD\) make the framework rigid. A weight of 10 lb. is attached to \(A\) and the framework is suspended from \(C\); find the stresses in all the rods of the system.