The straight line \(\frac{x-h}{\cos\alpha} = \frac{y-k}{\sin\alpha}\) through the point \(P\), whose coordinates are \((h,k)\), meets the parabola \(y^2=4ax\) in the points \(Q_1, Q_2\). Obtain a quadratic equation whose roots are the lengths of \(PQ_1, PQ_2\). If \(P\) is on the parabola prove that the length of \(Q_1Q_2\) is \[ \frac{4a\sin(\psi-\alpha)}{\sin^2\alpha\sin\psi}, \] where \(\psi\) is the inclination of the tangent at \(P\) to the axis of \(x\).
Find the coordinates of the point of intersection of the normals to the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at two points whose eccentric angles are \(\phi_1\) and \(\phi_2\). Prove that the normals at the two points in which the ellipse is cut by the straight line \[ \frac{x\cos\phi}{a^3} + \frac{y\sin\phi}{b^3} + \frac{1}{a^2-b^2}=0 \] meet on the ellipse at the point whose eccentric angle is \(\phi\).
Prove that the conic, whose equation in areal coordinates is \[ lx^2+my^2+nz^2+2pyz+2qzx+2rxy=0, \] is a rectangular hyperbola if \[ (m+n-2p)\cot A + (n+l-2q)\cot B + (l+m-2r)\cot C = 0. \] Shew that \((x-y-z)^2+4xz(1+\tan B\cot C)=0\) is a rectangular hyperbola and find the coordinates of its centre.
If \(y = \sqrt{1-x^2}.\sin^{-1}x\), prove that
Find the equations of the tangent and normal at the point \((h,k)\) of the curve whose equation is \(4x^3=27ay^2\). Shew that coordinates of the centre of curvature at the point \((h,k)\) are \[ \left(-h-\frac{h^2}{2a}, 4k+\frac{9ak}{h}\right). \]
Evaluate \[ \int (1+x)\sqrt{1-x^2}dx, \quad \int_0^\pi \cos 2\theta \log(1+\tan\theta)d\theta, \quad \int_0^{\pi/2} \sin^2\theta\cos 2\theta d\theta. \] Prove that \[ (m+np)\int x^{m-1}(x^n+a^n)^p dx = x^m(x^n+a^n)^p + npa^n\int x^{m-1}(x^n+a^n)^{p-1}dx. \]
Prove that the area of the loop of the curve \(y^2(a+x)=x^2(a-x)\) is \(2a^2(1-\frac{\pi}{4})\) and that the volume formed by the revolution of the loop about the axis of \(x\) is \(2\pi a^3(\log 2 - \frac{2}{3})\).
Two rough planes inclined to the horizontal at angles \(\alpha\) intersect in a horizontal line, forming a V. A thin rod, whose centre of gravity divides it into segments of lengths \(l_1\) and \(l_2\), rests in equilibrium with its extremities on the planes, the rod being perpendicular to their line of intersection and making an angle \(\theta\) with the horizontal. The angle \(\theta\) is measured positive when the portion \(l_2\) is higher than the portion \(l_1\). Show that if \(\lambda < \alpha\), where \(\lambda\) is the angle of friction, the angle \(\theta\) must lie between the values given by the equations \[ \tan\theta = (\sin^2\alpha-\sin^2\lambda)^{-1/2}\left[\frac{l_1-l_2}{l_1+l_2}\sin\alpha\cos\alpha \pm \sin\lambda\cos\lambda\right], \] but that if \(\lambda \ge \alpha\) equilibrium is possible in any position.
A given line \(L\) is perpendicular to a given force \(P\) and to the axis of a given couple \(G\). Show that the system consisting of \(P\) and \(G\) may in general be resolved into a force along \(L\) and another force, and give a construction for the point of intersection of the second force and the plane through \(P\) perpendicular to \(L\). Hence or otherwise show that given any line \(L\) and a system of forces which does not reduce to a single force or a couple, the system may be resolved uniquely into a non-zero force along \(L\) and a second force, unless \(L\) is parallel to the central axis or the moment of the system about \(L\) is zero, in which cases the resolution is impossible. Show further that if the system reduces to a single force or a couple, the resolution is impossible unless \(L\) is coplanar with the force or parallel to the plane of the couple, in which cases the resolution is possible in an infinity of ways.
A particle is projected from a point \(O\) at an angle \(\phi\) with the horizontal in a medium which causes a resistance proportional to the velocity. It meets the horizontal plane through \(O\) in a point \(P\), and at \(P\) the direction of motion of the particle makes an angle \(\omega\) with the horizontal. Prove that when \(\phi\) is such that \(OP\) is a maximum for a given velocity of projection, the values of \(\phi\) and \(\omega\) are connected by the relation \[ \phi+\omega = \frac{1}{2}\pi. \] Deduce that for this trajectory \(\phi < \frac{1}{4}\pi\).