Find the respective conditions that the line \(lx+my+n=0\) (1) touches, (2) is normal to the parabola, \(y^2-4ax=0\). Two parabolas touch one another at a common vertex: prove that a chord of one parabola, which lies along a tangent to the second, is divided in a constant ratio by the common axis.
Deduce the equation \(x^2/a^2+y^2/b^2=1\) of an ellipse from the definition that it is the locus of a point \(P\), such that \(SP+PH\) is constant, where \(S\) and \(H\) are the foci. Prove that the equation of the locus of the centre of the circle inscribed in the triangle \(SPH\) is \[ (1-e)x^2 + (1+e)y^2 = a^2 e^2 (1-e). \]
Prove that the line \(lx+my+n=0\) touches the conic \(Ax^2+2Hxy+By^2=1\), provided \(Am^2 - 2Hlm + Bl^2 = (AB-H^2)n^2\). Prove that the two conics, \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad \text{and} \quad -\frac{x^2}{a^2} - \frac{y^2}{b^2} + 2\mu\frac{xy}{ab} = (1+\mu^2)\frac{a^2+b^2}{a^2-b^2}, \] are such that any common tangent terminated by the points of contact subtends a right angle at the common centre.
Show that the conics which touch four given straight lines have their centres on a straight line. A family of conics has one focus and two tangents given: prove that the auxiliary circles and the director circles form two coaxal systems of circles, of which the respective radical axes are parallel.
Prove that two conics have four common points and four common tangents, and deduce that the relation between \(r\) and \(p\) for any conic, where \(r\) is the distance of a point on the conic from a chosen origin and \(p\) the perpendicular from the origin on the tangent, is of the fourth degree in \(r\) and in \(p\). In the case of the parabola \(y^2-4ax=0\) with the new origin at \(x=a+h, y=0\), prove that the \(p\) and \(r\) equation is \((aR-hP)^2=P^3 R\), where \(R=r^2-4ah, P=p^2-4ah\).
Two triangles \(ABC, A'B'C'\) are such that lines through \(A,B,C\) parallel respectively to \(B'C', C'A', A'B'\) are concurrent. Shew that the same is true of lines through \(A', B', C'\) parallel to \(BC, CA, AB\).
Given four points and one line, shew that there is in general one and only one conic through the four points which has an axis parallel to the given line; and give a geometrical construction for any number of points upon it. Indicate the case in which there are an infinite number of conics satisfying the given conditions.
Shew that there are in general two triangles whose sides pass through three given points and whose vertices lie on a given conic.
Two circles intersect orthogonally in two fixed points. Shew that their common tangent envelopes an ellipse of eccentricity \(1/\sqrt{2}\).
Two of the normals from a point \(P\) to a given parabola make equal angles with a given straight line. Prove that the locus of \(P\) is a parabola.