Prove that the locus of the feet of perpendiculars from a focus to tangents to a conic is a circle. \par The tangent to this circle at \(Y\) meets the major axis of the conic in \(R\), and tangents to the conic from \(Y\) touch at \(P\) and \(Q\); shew that \(P, Q, R\) are collinear.
Shew that the reciprocal of a circle with respect to a circle is a conic. \par Reciprocate the following theorem with respect to any point:-- \par Two circles cut at \(A, B\), and a line touches them at \(C, D\). \(CA\) cuts one circle in \(E\) and \(DB\) cuts the other in \(F\). Then \(CF\) and \(DE\) are parallel.
Prove the constant cross ratio property of four points of a conic. \par \(ABC\) is a triangle and \(P\) any point on the circumcircle. \(BP\) meets \(AC\) in \(M\) and \(CP\) meets \(AB\) in \(N\). Shew that \(MN\) envelops a conic.
The equation of two lines is \(ax^2+2hxy+by^2=0\); find the equation of the lines bisecting the angle between them. \par If \(lx+my=1\) bisects the angle between two lines one of which is \(px+qy=1\), shew that the other line is \[ (px+qy-1)(l^2+m^2) = 2(pl+qm)(lx+my-1). \]
Find the radical axis of the circles \[ x^2+y^2-4x-2y+4=0, \quad x^2+y^2+4x+2y-4=0, \] and the coordinates of the limiting points of the coaxal system to which they belong. \par Find the equation of the circle which touches them externally and passes through the point (1, 2).
Shew that the normal to a parabola at the point \(x=am^2, y=2am\) is \(y+mx=2am+am^3\). \par The tangent at \(P\) to the above parabola makes an angle \(\psi\) with the axis of the curve. The normal at \(P\) meets the curve at \(Q\). Shew that \(PQ\) is equal to \(4a/\sin\psi\cos^2\psi\) and that the least value of \(PQ\) is \(6a\sqrt{3}\).
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 \[ \frac{x^2+y^2+z^2-a^2}{x} = \frac{x^2+y^2+z^2-b^2}{y} = \frac{x^2+y^2+z^2-c^2}{z} = -(x+y+z). \]