Find the condition that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] may represent two straight lines. Assuming the condition to be satisfied, prove that the product of the distances of the two lines from the origin is \[ c/\sqrt{\{(a-b)^2+4h^2\}}. \]
Write down the equations of the tangent and normal at the point \((am^2, 2am)\) on the parabola \(y^2=4ax\). The normals at \(A,B,C\) to the parabola meet in a point \((5a, k)\). Prove that the orthocentre of the triangle \(ABC\) lies on the directrix.
Trace the curve \[ 13x^2-6xy+5y^2-4x-12y-4=0 \] and find its foci.
Prove that the equations \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \quad \frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{a^2-b^2}{a^2+b^2} \] represent confocal conics, and shew that every conic through the ends of the axes of the ellipse cuts the hyperbola orthogonally.
Prove the harmonic property of the pole and polar with respect to a conic. Two points \(P, P'\) are conjugate with regard to an ellipse and \(T\) is any point on the curve. \(PT, P'T\) meet the curve again in \(Q, Q'\). Prove that \(QQ'\) passes through the pole of \(PP'\).
If \(S=0\) represents a conic and \(\alpha=0\) a straight line, what locus does \(S-k\alpha^2=0\) represent? Two conics \(U, V\) each have double contact with a conic \(W\). Prove that the common chords of \(U, W\) and \(V, W\) pass through the intersection of a pair of the common chords of \(U, V\) and that these four common chords form a harmonic pencil.
Through a given point \(O\) draw three straight lines \(OA, OB, OC\) of given lengths so that \(A,B,C\) may be collinear and \(AB=BC\).
Define the radical axis of two circles. Given two circles \(A, B\) and a straight line \(L\), draw a circle \(C\) touching \(B\) and so that \(L\) is the radical axis of \(A\) and \(C\). Discuss the different cases.
Prove that the inverse of a circle is a circle or a straight line. Find the locus of points from which two equal intersecting circles invert into two equal circles.
Define the polar of a point with respect to a circle and shew that a straight line through a point cutting a circle is divided harmonically by the circle, the point and the polar of the point. A quadrilateral is inscribed in a circle and tangents are drawn at the angular points forming another quadrilateral. Prove that the third diagonals of these two quadrilaterals are coincident, and the angular points of one are harmonically conjugate to the corresponding angular points of the other.