Analytic Geometry Answers

By a suitable change of origin, show that the equation 9(x − 5)² + 16(y + 3)² = 144 represent an ellipse and that the coordinates of the centre are (5, −3). Find the position vectors of the foci and the ends of the major and minor axes.

Write down the parametric equations of the following ellipses. Find their eccentricities, foci and directrices. (a) 4x² + 9y2 = 4 (b) x²/9 + y²/16 = 1 (c) x²/7 + y²/14 = 1

 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.