Analytic Geometry Answers

 Prove that the line x − 2y + 4a = 0 touches the parabola y²= 4ax, and find the coordinates of P, the point of contact. If the line x − 2y + 2a = 0 meets the parabola in Q, R, and M is the mid-point of QR, prove that PM is parallel to the axis of x, and that this axis and the line through M perpendicular to it meet on the normal at P to the parabola.

The coordinates of the ends of a focal chord of the parabola y²= 4ax are (x₁, y₁) and (x₂, y₂). Show that x₁x₂ = a² and y₁y₂ = −4a².

The tangents at the points P(ap² , 2ap) and Q(aq² , 2aq) on the parabola y²= 4ax intersect at the point R. Given that the tangent at P is perpendicular to the chord OQ, where O is the origin, find the equation of the locus of R as p varies.

The points P(ap² , 2ap) and Q(aq² , 2aq) lie on the parabola y²= 4ax. Prove that if PQ is a focal chord then the tangents to the curve at P and Q intersect at right angles at a point on the directrix.

 Show that, if the chord joining the points P(ap² , 2ap), Q(aq² , 2aq) on the parabola y²= 4ax passes through (a, 0), then pq = −1. Further, the tangent at P meets the line through Q parallel to the axis of the parabola at R. Prove that the line x + a = 0 bisects PR.

The normal to the parabola y²= 4ax at the point P(at² , 2at) meets the x-axis at A. Find the equation of the locus of the midpoint of AP as t varies.

The tangent to the parabola y²= 4ax at the point P(at² , 2at) meets the x-axis at A and the y-axis at B. Find the equation of the locus of the mid-point of AB at t varies.

 Show that the line x + y − 2 = 0 is a tangent to the parabola x²+ 8y = 0 and find the position vector of the point of contact.

The lines joining the origin to the points of intersection of the line lx + my = 1 and y²= 4ax are at right angles. Show that 4al = 1.

If the position vector of one end of a chord through the focus of the parabola y²= 8x is 1/2i + 2j, find the position vector of the other end.