Conic sections

The ellipse E has equation

The line with equation y = mx + 4 is a tangent to E

Without using differentiation show that m = ±

The curve C₁ has equation

The curve C₂ has equation

Find a sequence of transformations that maps the graph of C₁ onto the graph of C₂

Find the equations of the asymptotes to C₂

Give your answers in the form ax by c 0 where a, b and c are integers.

The ellipse E₁ has equation

Write down the equation of E₂

The ellipse E₃ has equation

Describe the transformation that maps E₂ to E₃

Explain, without doing any calculations, why L_A has an equation of the form

where c is a constant.

H₁ is the locus of points such that the distance from the point (5, 0) is twice the distance from the line x 2

Show that the equation of H₁ can be written in the form

where q and r are integers.

H₂ is the hyperbola

Describe fully a sequence of two transformations which maps the graph of H₂ onto the graph of H₁