3 Further transformations of the ellipse, hyperbola and parabola 6
a Sketch the curve C1 with equation y2 = 5x. The curve C2 is an enlargement of C1 with scale factor 3 (and centre at the origin). b Find the equation of C2. c Describe a horizontal stretch that transforms C2 back to C1.
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2 y2 Describe fully the transformation that maps the hyperbola x − = 1 onto the hyperbola 5x2 − 3y2 = 105. 3 5
Describe fully a possible transformation that maps the curve x2 − 3y2 = 3 onto the curve y2 − 3x2 = 3. Hence state the equations of the asymptotes of the curve y2 − 3x2 = 3. 2 y2 a Sketch the hyperbola with equation x − = 1. 3 12 b Show that an enlargement with centre at the origin does not change the equations of the asymptotes.
y2 x2 − = 1 is a hyperbola and find the equations of its asymptotes. 9 16
pl e
10 Show that the curve with equation
Section 3: Combined transformations
Rewind
You can combine any of the transformations from Sections 1 and 2. You know that sometimes the order of transformations matters – performing transformations in a different order may produce a different curve.
Sa m
WORKED EXAMPLE 3.6
You met combined transformations in A Level Mathematics Student Book 2, Chapter 3.
2 y2 Find the equation of the ellipse x + = 1 after each sequence of transformations. 16 9 −1 a Enlargement with scale factor 3 followed by a translation with vector . 4 −1 b Translation with vector followed by an enlargement with scale factor 3. 4
( )
a After the enlargement:
() ()
y 2 x 2 3 3 + =1 9 16 2 2 y ⇔ x + =1 144 81
After the translation: (x + 1) 2 ( y − 4) 2 + =1 144 81 b After the translation:
( )
y The enlargement replaces x by x and y by . 3 3 Remember that both variables are squared.
Replace x by x + 1 and y by y − 4.
Now do the translation first.
(y − 4) (x + 1) + =1 9 16 2
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