Problems

Find the equation of the pair of tangents drawn from a given point to the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0. \] Two tangents are drawn from a point \(P\) to a conic to meet a fixed tangent at \(Q\) in \(T\) and \(T'\). Shew that if \(QT \cdot QT'\) is constant the locus of \(P\) is a straight line parallel to the tangent at \(Q\).

A point on the conic \(y^2=kxz\) is given by the parameter \(\lambda\) where \(x=\lambda^2y\); prove that the equation of the line joining the points \(\lambda\) and \(\lambda'\) is \[ x-y(\lambda+\lambda') + k\lambda\lambda'z = 0. \] Each of two sides of a triangle inscribed in a conic passes through a fixed point; prove that the third side always touches a conic having double contact with the given one.

Factorise

1. [(i)] \(a^3(b-c)+b^3(c-a)+c^3(a-b)\),
2. [(ii)] \((bc-a^2)(ca-b^2)+(ca-b^2)(ab-c^2)+(ab-c^2)(bc-a^2)\).

Investigate the conditions that \(ax^2+2bx+c\) should be positive for all real values of \(x\). Prove that the expression \(\frac{x+a}{x^2+bx+c^2}\) will be capable of all real values if \(a^2+c^2 < ab\).

Find the sum of the squares, and the sum of the cubes of the first \(n\) natural numbers. Sum the series to \(n\) terms

1. [(i)] \(1+3^2x+5^2x^2+\dots\);
2. [(ii)] \(m!+(m+1)!/1!+(m+2)!/2!+\dots\).

Prove the law of formation of successive convergents of the continued fraction \[ a_1 + \frac{1}{a_2+} \frac{1}{a_3+} \frac{1}{a_4+} \dots. \] Prove that the difference between the continued fraction and \(p_n/q_n\) its \(n\)th convergent is less than \(1/q_n q_{n+1}\) and greater than \(a_{n+2}/q_n q_{n+2}\).

Find the differential coefficients of \(f(x)/\phi(x)\) and of \(f\{\phi(x)\}\). Find the \(n\)th differential coefficients of \(\frac{x}{x^2-3x+2}\) and of \(\tan^{-1}x\).

Prove the method of determining and discriminating between maximum and minimum values of a function of a single variable by means of its differential coefficients. If \(O\) be the centre of an ellipse whose semi-axes are \(a\) and \(b\), \(ON\) the perpendicular to the tangent at \(P\), shew that the maximum area of the triangle \(OPN\) is \(\frac{1}{4}(a^2-b^2)\).

Investigate the equation of the tangent at any point of the curve \(f(x,y)=0\). Write down the equation of the normal at the point of the curve \(ay^2=x^3\) where \(x=\frac{1}{2}a\), and prove that it touches the curve.

Prove the expressions for the radius of curvature of a curve \[ \text{(i) } \rho = r\frac{dr}{dp}, \quad \text{(ii) } \rho = p + \frac{d^2p}{d\psi^2}. \] Find \(\rho\) and \(p\) at a point of \(r^n=a^n\sin n\theta\).