Find the locus of the centres of circles passing through a given point and cutting a given circle orthogonally.
\(A\) and \(B\) are two fixed points and \(\lambda\) is a fixed line through \(A\); a variable circle through \(A\) and \(B\) cuts \(\lambda\) again in \(P\). Prove that the tangent at \(P\) to this circle touches a fixed parabola with its focus at \(B\).
\(ABC\) is a triangle inscribed in a conic and the points \(Q\) and \(R\) on \(CA\) and \(AB\) respectively are conjugate with respect to the conic; prove that the lines \(QR\) and \(BC\) are conjugate with respect to the conic.
\(A\) and \(B\) are two fixed points and \(\lambda\) and \(\mu\) are two fixed lines in a plane; prove that the locus of a point \(P\), such that \(PA, PB\) are harmonically separated by the lines through \(P\) parallel to \(\lambda\) and \(\mu\), is a hyperbola, whose asymptotes are the lines through the middle point of \(AB\) parallel to \(\lambda\) and \(\mu\).
State and prove the harmonic property of a quadrangle. If \(L, M, N\) are the feet of the perpendiculars from the vertices \(A, B, C\) of a triangle to the opposite sides, prove that the triangle \(LMN\) is self polar with respect to any rectangular hyperbola through \(A, B, C\).
Prove that the equation of the circumcircle of the triangle whose sides lie along the lines \(ax^2+2hxy+by^2=0\), \(lx+my+n=0\) is \[ (am^2-2hlm+bl^2)(x^2+y^2) + n(a-b)(my-lx) - 2hn(mx+ly)=0. \] Interpret this result geometrically, when \(am^2-2hlm+bl^2=0\).
The sides of a variable triangle with its centroid at the fixed point \((x_1, y_1)\) touch the parabola \(y^2=4ax\); prove that the vertices of the triangle lie on the parabola \[ 2(y^2-ax)-3y_1y+6ax_1=0. \]
Prove that the condition that the line \(lx+my+n=0\) should touch the parabola, whose focus is \((\alpha, \beta)\) and whose directrix is \(px+qy+r=0\), is \[ (p\alpha+q\beta+r)(l^2+m^2) - 2(pl+qm)(\alpha l + \beta m + n) = 0. \]
Prove that the locus of the poles of a given straight line with respect to a system of confocal conics is a straight line. If the given straight line is \(lx+my+n=0\) and one of the conics is \(ax^2+2hxy+by^2+c=0\), prove that the locus of poles is the line \[ n(ab-h^2)(mx-ly) + c\{(a-b)lm - h(l^2-m^2)\} = 0. \] Interpret this result geometrically, when (i) \((a-b)lm-h(l^2-m^2)=0\), (ii) \(ab-h^2=0\).
A variable conic touches a fixed line and also touches the sides of a fixed triangle; prove that for any such conic the lines joining the vertices of the triangle to the points of contact of the opposite sides meet in a point \(P\) and that, as the conic varies, the locus of \(P\) is a conic circumscribing the fixed triangle.