Ellipses are drawn through the middle points of the sides of the rectangle \((x^2-a^2)(y^2-b^2)=0\). Find the general equation of the family; and shew that they are all cut four times orthogonally by one of the hyperbolas having the diagonals as asymptotes.
Equal circles of radius \(r\) have their centres at the points \((\pm a, 0)\). Shew that tangents drawn to them from any point on the conic \[ r^2(x^2-a^2)+(r^2-2a^2)(y^2-r^2)=0 \] form a harmonic pencil. \par Examine the special cases when the circles (i) touch, (ii) cut orthogonally.
Find the general equation of all pairs of lines having the same angle-bisectors as \(ax^2+2hxy+by^2=0\). \par Shew that the general equation of any conic confocal with \(ax^2+2hxy+by^2+c=0\) may be written in the form \[ (ax^2+2hxy+by^2)+\lambda(x^2+y^2)+c\frac{(a+\lambda)(b+\lambda)-h^2}{ab-h^2}=0. \]
Find in areal coordinates, referred to a triangle with sides \(a, b, c\), the equation of the conic which touches the sides at their middle points; and verify that the centre of the conic is the centroid of the triangle. \par Shew that the equation of the axes of the conic may be written \[ (b^2-c^2)(y-z)^2+(c^2-a^2)(z-x)^2+(a^2-b^2)(x-y)^2=0. \]
If \[ (1+3\sin^2\phi)^{\frac{1}{3}} = \sin^{\frac{2}{3}}\theta + \cos^{\frac{2}{3}}\theta, \] prove that \[ (1+3\tan^2\phi)\tan\theta = 2\tan^3\phi. \]
An attempt is made to construct a right angle by means of three strings of lengths 3, 4 and 5 yards. If the third string is one inch too long, find the resulting error in the right angle in minutes of arc.
Expand \(\frac{\sin n\theta}{\sin\theta}\) in a series of descending powers of \(\cos\theta\), when \(n\) is an odd integer. \par Prove also that when \(n\) is odd \[ n\operatorname{cosec} n\theta = \operatorname{cosec}\theta + \operatorname{cosec}\left(\theta+\frac{2\pi}{n}\right) + \operatorname{cosec}\left(\theta+\frac{4\pi}{n}\right) + \dots \text{ to } n \text{ terms}. \]
Prove that in areal coordinates the equation of the circumcircle of the triangle of reference is \(a^2yz+b^2zx+c^2xy=0\), and that the polar reciprocal of this circle with respect to the conic \(ux^2+vy^2+wz^2=0\) is \[ a\sqrt{(ux)}+b\sqrt{(vy)}+c\sqrt{(wz)}=0. \]
Find from the definition the derivative of \(\sin^{-1}x\). \par Prove that for the value \(x=0\), \(\frac{d^n}{dx^n}(\sin^{-1}x)=(1,3,5,\dots n-2)^2\) or 0 according as \(n\) is odd or even.
Find the maximum and minimum values of \(y=(x+1)^2(x+3)^3(x+2)\) and draw a rough graph of the curve.