Conic sections

The ellipse E has equation

The line l has equationy = mx + cwhere m and c are constants.

Show that the x coordinates of the points of intersection of E and l satisfy the equation

where k is a constant to be determined.

Hence, given that the line l

• is a tangent to E

• passes through the point (10, 20)

determine the possible equations of l

The ellipse E has equation

The hyperbola H has equation

where a and b are positive constants.

Given that

• the eccentricity of H is the reciprocal of the eccentricity of E

• the coordinates of the foci of H are the same as the coordinates of the foci of E

determine

the value of a

the value of b

The ellipse E has equation

Determine the exact value of the eccentricity of E

The points P (4 cosθ, 3 sinθ) and Q (4 cosθ, –3 sinθ) lie on E where 0 < θ < The line l₁ is the normal to E at the point P

Use calculus to show that l₁ has equation

4 x sin θ – 3 y cos θ = 7 sin θ cos θ

The line l₂ passes through the origin and the point Q The lines l₁ and l₂ intersect at the point R

Determine, in simplest form, the coordinates of R

Hence show that, as θ varies, R lies on an ellipse which has the same eccentricity as ellipse E

An ellipse has equation 1 and eccentricity e₁ A hyperbola has equation 1 and eccentricity e₂

Given that e₁ e₂ = 1

show that a² = 3b²

Given also that the coordinates of the foci of the ellipse are the same as the coordinates of the foci of the hyperbola,

determine the equation of the hyperbola.

The rectangular hyperbola H has equation xy = 36

Use calculus to show that the equation of the tangent to H at the point P 6t is

The point Q 12t also lies on H.

Find the equation of the tangent to H at the point Q.

The tangent at P and the tangent at Q meet at the point R.

Show that as t varies the locus of R is also a rectangular hyperbola.

The ellipse E has equation

Find

the coordinates of the foci of E,

the equations of the directrices of E.

The parabola C has equation

and the hyperbola H has equation

Write down the equations of the asymptotes of H.

The line l₁ is normal to C and parallel to the asymptote of H with positive gradient. The line l₂ is normal to C and parallel to the asymptote of H with negative gradient.

Determine

an equation for l₁

an equation for l₂

The lines l₁ and l₂ meet H at the points P and Q respectively.

Find the area of the triangle OPQ, where O is the origin.

The ellipse E has equation

The points S and S′ are the foci of E.

Find the coordinates of S and S′

Show that for any point P on E, the triangle PSS′ has constant perimeter and determine its value.

The points P(9p², 18p) and Q(9q², 18q), p q, lie on the parabola C with equation

The line l passes through the points P and Q

Show that an equation for the line l is

(p + q) y = 2(x + 9 pq)

The normal to C at P and the normal to C at Q meet at the point A.

Show that the coordinates of A are

Given that the points P and Q vary such that l always passes through the point (12, 0)

find, in the form y²= f (x), an equation for the locus of A, giving f (x) in simplest form.

The parabola C has equation

The distinct points P(p², 4p) and Q(q², 4q) lie on C, where pq

The tangent to C at P and the tangent to C at Q meet at the point R (–28, 6).

Show that the area of triangle PQR is 1331

The hyperbola H has equation

The line l₁ is the tangent to H at the point P(4cosh , 3sinh ).

The line l₁ meets the x-axis at the point A.

The line l₂ is the tangent to H at the point (4, 0).

The lines l₁ and l₂ meet at the point B and the midpoint of AB is the point M.

Show that, as θ varies, a Cartesian equation for the locus of M is

where p and q are values to be determined.

Let S be the focus of H that lies on the positive x-axis.

Show that the distance from M to S is greater than 1