If \(\lambda = \frac{L_1 x^2 + 2M_1 x + N_1}{L_2 x^2 + 2M_2 x + N_2}\), prove that the condition for \(\lambda\) to attain all real values if \(x\) assumes all real values is \[ (L_1 N_2 - N_1 L_2)^2 < 4(M_1 N_2 - N_1 M_2)(L_1 M_2 - M_1 L_2). \] Find the limitations on the value of \(\lambda\) if this condition is not satisfied. Shew that the turning values of \(\frac{a_1 x^2+b_1}{a_2 x^2+b_2}\) are \(\frac{a_1}{a_2}, \frac{b_1}{b_2}\).
Two men of height \(d\) feet (to the level of the eyes) are walking on the same horizontal level round a conical hill at a depth \(h\) feet below the summit. The total height of the hill is 300 feet and the diameter of its base 800 feet. When the man in front has just disappeared from sight round the hill, shew that the shortest distance \(S\) between them over the hill is given by \[ 5\sin^{-1}\left(\frac{3S}{10h}\right) = 4\cos^{-1}\left(\frac{h-d}{h}\right). \]
\(V\) is the middle point of a given chord \(AB\) of a given circle. \(PQ\) is any parallel chord. \(QV\) meets the circle again in \(R\). Prove that \(PR\) passes through a fixed point.
\(PQ\) is a chord of a parabola that passes through the focus \(S\). Two circles are drawn through \(S\), touching the parabola at \(P\) and \(Q\) respectively. Prove that the circles cut each other orthogonally.
If the tangent at \(P\) to an ellipse meets a directrix in \(R\), and if \(S\) is the corresponding focus, prove that \(PSR\) is a right angle. If two conjugate diameters of an ellipse are produced to meet a directrix, prove that the orthocentre of the triangle so formed is the corresponding focus.
From the point \(Q\) in which the tangent at any point \(P\) of a hyperbola meets an asymptote, perpendiculars \(QM, QN\) are drawn to the axes. Prove that \(MN\) passes through \(P\).
Any number of spheres touch a plane at the same point \(O\). Prove that any plane, not through \(O\), cuts them in a system of co-axal circles.
A variable chord \(PQ\) of a given circle subtends a right angle at a given point \(A\). Find the locus of the pole of \(PQ\) with respect to the circle. Interpret the result when \(A\) lies on the given circle.
The tangent at any point \(P\) of the parabola \(y^2=4ax\) is met in \(Q\) by a line through the vertex \(A\) at right angles to \(AP\), and \(Z\) is the foot of the perpendicular from \(A\) as the tangent at \(P\). Shew that there are three positions of the point \(P\) on the parabola for which \(Z\) lies on the straight line \(lx+my+na=0\), and that the corresponding points \(Q\) lie on the line \((2l-n)x+4my+2na=0\).
Prove that if the normals at four points of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) are concurrent, and two of the points lie on the line \(\frac{lx}{a}+\frac{my}{b}+1=0\), the other two will lie on the line \(\frac{x}{al}+\frac{y}{bm}+1=0\). Prove also that if two lines drawn through the point \((\frac{2}{3}a, \frac{2}{3}b)\) meet the ellipse at four points, the normals at which are concurrent, one of the lines is \(\frac{x}{a}-\frac{y}{b}=2\), and find the equation of the other.