A smooth parabolic tube is fixed in a vertical plane with its vertex downwards. A particle starts from rest at the extremity of the latus rectum (\(4a\) in length), and slides down the tube; express as a definite integral the time taken to reach the vertex, and shew that this time is approximately \(2.7 \sqrt{\frac{a}{g}}\) seconds.
Prove that the feet of the perpendiculars drawn on the sides of a triangle from any point of the circumcircle are collinear. If \(O\) is a point on a circle the circles on the chords \(OP, OQ, OR\) as diameters intersect in three points other than \(O\) which are collinear.
The tangent at \(P\) of a central conic meets the minor axis in \(L\). Shew that the angle \(LSP\) is equal to the angle made by the tangent with the major axis.
If a square number has 01 for its last two digits, the preceding digit will be even. Find the lowest number (other than 1) whose square ends with the digits 0001.
Prove that, in a triangle \(ABC\), \[ \Sigma \sin^2 A \tan A = \tan A \tan B \tan C - 2\sin A \sin B \sin C. \]
Prove graphically that the equation \(\theta=\cos\theta\) has only one real root, and that it is given approximately by \(\cos\theta = \cdot 8\).
Find the equations of two circles, which have \(x=a\) for radical axis and \((\pm c, 0)\) for centres of similitude, and one of which passes through the origin. Find the limiting points when \(a>c\).
State the chief properties of poles and polars with regard to a conic. A conic passes through the origin, and the polars of the points \((c,0), (0,c)\) are respectively \(mx+y+c=0, x+ny+c=0\). Prove that the polar of \((c,c)\) is \(mx+ny+2c=0\).
A triangle \(ABC\) is inscribed in an ellipse, and \(D\) is the pole of \(BC\). Prove that \(AD\) and the tangent at \(A\) are harmonically conjugate with \(AB, AC\). The tangent at \(P\) meets the directrices in \(L\) and \(M\), and the second tangents drawn from \(L, M\) intersect in \(Q\). Prove that \(Q\) lies on the normal at \(P\), and that the middle point of \(PQ\) is on the minor axis.
Write down the general equation of conics which touch the conic \(S=0\) at a given point. Find the equation of the rectangular hyperbola which touches the parabola \(y^2-4ax=0\) at the vertex, and intersects it at the points \((a, 2a), (4a, 4a)\). Prove that its asymptotes are \[ 10x+5y+12a=0, \quad 5x-10y+24a=0. \]