Find the latus rectum, equation of the axis, and the coordinates of the focus of \[ x^2+4xy+4y^2+4x-17y+19=0. \]
Prove that the feet of the perpendiculars from any point on the circumcircle of a triangle \(ABC\) on to the sides of the triangle are collinear. Prove also that the pedal lines of the extremities of a diameter of the circumcircle of the triangle are at right angles.
The tangents at \(P\) and \(Q\) to a parabola whose focus is \(S\), intersect at \(T\). Prove that the triangles \(SPT, STQ\) are similar. Any number of parabolas touch a given line and have a common focus; prove that their vertices all lie on a circle.
Prove that the length of that chord of the circle of curvature at a point \(P\) of an ellipse, which passes through the centre of the ellipse, is \[ \frac{2CD^2}{CP}. \] If \(S, H\) are the foci and \(B\) an extremity of the minor axis, prove that the circle \(SHB\) will cut the minor axis in the centre of curvature at \(B\).
If two circles cut orthogonally, prove that any diameter of either is cut harmonically by the other circle. \(ABCD\) is a cyclic quadrilateral, and \(EF\) is the third diagonal. Prove that the circle on \(EF\) as diameter cuts the circle \(ABCD\) orthogonally.
Prove that the sum of all the plane angles forming any solid angle is less than four right angles. Two walls of a house meet at right angles, and the roofs above these walls are inclined at 30° to the horizontal. Prove that their line of intersection is inclined to the horizontal at an angle \(\cot^{-1}\sqrt{6}\).
Find the equation of the polar of a point with respect to the circle \[ x^2+y^2=a^2. \] Circles are drawn through the given point \((c,0)\) touching this circle. Prove that the locus of the pole of the axis of \(x\) with respect to these circles is the curve \[ 4a^2[(x-c)^4 + x^2y^2] = y^2(a^2-c^2+2cx)^2. \]
If two normals to the parabola \(y^2=4ax\) make complementary angles with the axis, prove that their point of intersection lies on one of the parabolas \[ y^2=a(x-a), \] or \[ y^2=a(x-3a). \]
If \((h,k)\) is a point of intersection of the ellipses \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{and} \quad \frac{x^2}{a'^2} + \frac{y^2}{b'^2} = 1, \] prove that their common tangents are \[ \frac{hx}{aa'} \pm \frac{ky}{bb'} = 1, \] and that the product of the areas of the parallelograms formed by their four common points and their four common tangents is \[ 8aa'bb'. \]
Prove that the feet of the four normals from a point \(P\) to any central conic lie on a rectangular hyperbola which passes through the centre and through \(P\). If the conic is an ellipse, prove that the sum of the eccentric angles of the feet of the four normals is an odd multiple of \(\pi\).