Find the equation of the axes of the conic given by the general equation. Trace roughly the curve \(15x^2+40xy+24y^2-120x-120y=0\), and prove that its foci lie on the axes of coordinates.
Prove that of the circles \begin{align*} b(x^2+y^2) + a^2(2y-b) &= 0, \\ a(x^2+y^2) + b^2(2x-a) &= 0, \\ ab(x^2+y^2) + 2c^2(bx-ay) &= 0, \end{align*} the first two intersect at an angle of 120\(^\circ\); also that the third, if it is coaxial with the others, intersects each of them at 120\(^\circ\).
Determine the radius of curvature at any point of a curve whose coordinates are given in terms of a single parameter \(\theta\). The normal at any point \(P\) of the curve \(x=a\cos^3\theta, y=a\sin^3\theta\) meets the circle \(x^2+y^2=a^2\) in the points \(Q, R\). Prove that \(RP=3PQ=\rho\) the radius of curvature at \(P\). Also that, if \(s=0\) when \(\theta=\frac{1}{4}\pi\), \(16s^2+4p^2=9a^2\).
Find the coordinates of the double point of the cubic whose equation is \[ xy(5x+y-6)+3x+3y-2=0. \] Write down the equation of the tangents at the double point. Are they real?
Prove that, if \(\cos\beta = \cos\theta\cos\phi+\sin\theta\sin\phi\cos\alpha\), and \(\sin\alpha = e\sin\beta\) \[ d\theta\{1-e^2\sin^2\phi\}^{\frac{1}{2}} + d\phi\{1-e^2\sin^2\theta\}^{\frac{1}{2}} = 0. \]
Integrate with respect to \(\theta\) the expressions \(\frac{1}{\sin^3\theta}\) and \(\frac{5}{1+2\cot\theta}\). Prove that the straight line \(2a^2x=9b^2y\) cuts off from the curve \(b^2y=x^2(a-x)\) two segments which are equal in area.
A triangle moves so that each of two sides passes through a fixed point. Prove that its base touches a fixed circle.
\(TPT'\) is the tangent to a hyperbola, whose centre is \(C\), meeting the asymptotes in \(T\) and \(T'\). \(PQ\) parallel to \(CT\) meets the directrix in \(Q\). Prove that \(T'Q\) is parallel to \(ST\) where \(S\) is the focus inside the branch of the curve on which \(P\) lies.
Prove that a sphere can be drawn to cut orthogonally three circles in space, each of which intersects each of the other two in two points.
If \({}_nC_r\) is the coefficient of \(x^r\) in the expansion of \((1+x)^n\) by the binomial theorem where \(n\) is a positive integer, prove that \[ \sum_{s=0}^{s=n-r} {}_rC_s \cdot {}_{n-r}C_s = {}_{2n-r}C_n. \]