Coordinate Geometry & Conics

12 methods · 26 questions

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Whenever a STEP conic question talks about tangents, normals or chords at named points, never work with $(x, y)$ : assign each point its standard parameter and do everything in those parameters. A chord or tangent then has coefficients that are symmetric (or low-degree) in the parameters, which is what makes the later elimination clean.

Trigger: a conic plus the words tangent, normal, chord, or two/three points on the curve.

Instances:
(i) parabola $y^{2} = 4 a x$ as $(a t^{2}, 2 a t)$ : tangent $t y = x + a t^{2}$ , normal $y + t x = 2 a t + a t^{3}$ .
(ii) ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ as $(a cos α, b sin α)$ : tangent $\frac{x cos α}{a} + \frac{y sin α}{b} = 1$ .
(iii) hyperbola as $(a sec θ, b tan θ)$ , giving tangent $\frac{x sec θ}{a} - \frac{y tan θ}{b} = 1$ .
(iv) the rational (half-angle) parametrisation $(\frac{a ( 1 - t ^{2} )}{1 + t ^{2}}, \frac{2 b t}{1 + t ^{2}})$ when you want tangency to become a polynomial in $t$ .

Linked questions (3)

STEP 2 2006 · Q7

An ellipse has equation

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

. Show that the equation of the tangent at the point

(a cos α, b sin α)

y = - \frac{b cot α}{a} x + b cosec α .

The point

A

has coordinates

(- a, - b)

, where

a

and

b

are positive. The point

E

has coordinates

(- a, 0)

and the point

P

has coordinates

(a, k b)

, where

0 < k < 1

. The line through

E

parallel to

A P

meets the line

y = b

at the point

Q

. Show that the line

P Q

is tangent to the above ellipse at the point given by

tan (α /2) = k

.

Determine by means of sketches, or otherwise, whether this result holds also for

k = 0

and

k = 1

STEP 3 2008 · Q3

The point

P (a cos θ, b sin θ)

, where

a > b > 0

, lies on the ellipse

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 .

The point

S (- e a, 0)

, where

b^{2} = a^{2} (1 - e^{2})

, 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

S P

and

O N

intersect at

T

. Show that the

y

-coordinate of

T

\frac{b sin θ}{1 + e cos θ} .

Show that

T

lies on the circle with centre

S

and radius

a

STEP 3 2014 · Q3

The line $L$ has equation $y = m x + c$ , where $m > 0$ and $c > 0$ . Show that, in the case $m c > a > 0$ , the shortest distance between $L$ and the parabola $y^{2} = 4 a x$ is $\frac{m c - a}{m m ^{2} + 1} .$ What is the shortest distance in the case that $m c ⩽ a$ ?
Find the shortest distance between the point $(p, 0)$ , where $p > 0$ , and the parabola $y^{2} = 4 a x$ , where $a > 0$ , in the different cases that arise according to the value of $p / a$ . [You may wish to use the parametric coordinates $(a t^{2}, 2 a t)$ of points on the parabola.] Hence find the shortest distance between the circle $(x - p)^{2} + y^{2} = b^{2}$ , where $p > 0$ and $b > 0$ , and the parabola $y^{2} = 4 a x$ , where $a > 0$ , in the different cases that arise according to the values of $p$ , $a$ and $b$ .