Find the condition that the line \(lx+my=1\) may be a tangent to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), and shew that the locus of the point of intersection of perpendicular tangents to the ellipse is the circle \(x^2+y^2=a^2+b^2\). \par If \(P\) is any point on this circle, shew that the locus of the middle point of the chord in which the polar of \(P\) cuts the ellipse is \[ \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right)^2 = \frac{x^2+y^2}{a^2+b^2}. \]
Express the sum and the product of the squares of the semi-axes of the conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\) in terms of the coefficients. \par Find the equation of a conic through the origin having \(2x-3y+7=0\), \(3x+2y-6=0\) as axes and an asymptote parallel to the axis of \(x\).
Prove that, if \(y=(ax+b)/(cx+d)\), there are two values of \(x\) which are equal to the corresponding values of \(y\), and that these are real and distinct, coincident, or imaginary according as \((a+d)^2 >= \text{ or } < 4(ad-bc)\). Shew that, if these values are \(h, k\), the relation between \(y\) and \(x\) can be put in the form \(\frac{y-h}{y-k} = \lambda \frac{x-h}{x-k}\).
Solve the equations \[ \frac{x^2+y^2+z^2-a^2}{x} = \frac{x^2+y^2+z^2-b^2}{y} = \frac{x^2+y^2+z^2-c^2}{z} = -(x+y+z). \]
Sum the series
Prove the rule for forming the convergents to the continued fraction \[ a_1 + \frac{1}{a_2 +} \frac{1}{a_3 +} \dots, \] and prove that these convergents are fractions in their lowest terms. \par Prove that, if \(p/q\) is the fraction in its lowest terms, which is equal to \[ a_1 + \frac{1}{a_2 +} \frac{1}{a_3 +} \dots + \frac{1}{a_n +} \frac{1}{a_{n-1} +} \dots + \frac{1}{a_2 +} \frac{1}{a_1}, \] then \((q^2+1)/p\) is an integer when \(a_1, a_2, \dots a_n\) are all integers.
Prove that \[ 1-\cos^2 A - \cos^2 B - \cos^2 C + 2\cos A \cos B \cos C = 4 \sin S \sin(S-A) \sin(S-B) \sin(S-C) \] where \[ 2S = A+B+C. \] Given that \(a \sin\alpha = b\sin(\alpha+\beta) = c\sin(\alpha+2\beta)\) express the ratios \[ \cos\alpha:\cos(\alpha+\beta):\cos(\alpha+2\beta) \] in terms of \(a, b, c\).
In solving a triangle in which two sides and the included angle are given, shew how to determine the error introduced into the calculated value of the third side by small errors in the given quantities. \par In a triangle \(ABC\) we have given that approximately \[ a=36 \text{ ft.}, \quad b=50 \text{ ft.}, \quad C=\tan^{-1}\frac{3}{4}; \] find what error in the given value of \(a\) will cause an error in the calculated value of \(c\) equal to that caused by an error of \(5'\) in the measurement of \(C\).
Express \((a+b\sqrt{-1})^{c+d\sqrt{-1}}\) in the form \(A+B\sqrt{-1}\) where all the quantities \(a, b, \dots\) are real. \par Prove that \[ \sum_{r=1}^5 \tan^{-1}\left(\frac{1}{\alpha_r+1}\right) = \tan^{-1}\frac{4}{3}, \] where \(\alpha_1, \dots \alpha_5\) are the fifth roots of unity.
Differentiate \(x^{\log x}\), \((\log x)^x\). \par Find the \(n\)th differential coefficient of \(a^x \sin^n x\).