Embedded Files
Conics in Rectangular Form
STEP 2013, Math II, #4: The line passing through the point with gradient intersects the circle of unit radius centred at the origin at P and Q, and M is the midpoint of the chord PQ. Find the coordinates of M in terms of
and .
(i) Suppose
is fixed and positive. As varies, M traces out a curve (the locus of M). Show that on this curve. Given that varies with , show that the locus is a line segment of length
. Give a sketch showing the locus and the unit circle.
(ii) Find the locus of M in the following cases, giving in each case its cartesian equation, describing it geometrically and sketching it in relation to the unit circle.
(a)
is fixed with , and varies with ;
(b)
, and varies with .
STEP 2002, Math I, #1: Show that the equation of any circle passing through the points of intersection of the ellipse and the ellipse
can be written in the form
.
STEP 2005, Math I, #6: The point
has coordinates and the point has coordinates . The variable point has coordinates and moves on a
path such that
. Show that the Cartesian equation of the path of is
.
The point
has coordinates and the point has coordinates . The variable point moves on a path such that
where
. Given that the path of is the same as the path of , show that
.
Show further that , in the case .
Conics in Polar Form
STEP 2008, Math III, #3: The point , where , lies on the ellipse
The point
, where , is a focus of the ellipse. The point N is the foot of the perpendicular from the origin, O , to the tangent to the ellipse at P. The lines SP and ON intersect at T. Show that the y-coordinate of T is
Show that T lies on the circle with centre S and radius
.
Page updated
Google Sites
Report abuse