Find the limiting values of the expression \[ \frac{\sin x \sin y}{\cos x - \cos y} \] as the point \((x,y)\) approaches the origin along curves of the form (i) \(y=xk\), where \(k\) is positive, (ii) \(y=ax+bx^2\), where \(a\) and \(b\) have various constant values. Point out any cases in which the limits are infinite.
Integrate \[ (1+x^2)^{\frac{3}{2}}, \quad \frac{1-\tan x}{1+\tan x}. \] Prove that, if \(n\) is a positive integer, \[ \int_{-\infty}^{\infty} \frac{dx}{(1+x^2)^{n+1}} = \frac{1 \cdot 3 \dots (2n-1)}{2 \cdot 4 \dots 2n} \pi. \]
Prove that \[ \int_0^\pi \frac{\sin n\theta}{\sin\theta} d\theta = 0 \text{ or } \pi, \text{ according as } n \text{ is an even or odd positive integer.} \] Evaluate \(\displaystyle\int_0^\pi \frac{\sin^2 n\theta}{\sin^2\theta} d\theta\), where \(n\) is a positive integer.
A variable circle touches both a given circle and a given straight line. Prove that the chord of contact passes through a fixed point.
Prove that the inverse of a system of non-intersecting coaxal circles with respect to a limiting point is a system of concentric circles having the inverse of the other limiting point as centre. \(S_1, S_2\) are two circles and \(L\) is a limiting point of the system of which they are members. A circle drawn through \(L\) and touching \(S_1\) at \(X\) meets \(S_2\) in \(P\) and \(Q\). Prove that \[ \frac{PX}{QX} = \frac{PL}{QL}. \]
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 \[ y^2 - ax + 2a^2 = \pm a^2. \]