Tangents and normals

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 parabola P has equationy² = 4ax, where a is a positive constant. The point Aat², 2at, wheret 0, lies on P.

Use calculus to show that an equation of the tangent to P at A is

The point B(2k², 4k) and the point C(2k², –4k), where k is a constant, lie on P.

The tangent to P at B and the tangent to P at C intersect at the point D.

Given that the area of the triangle BCD is 432

determine the coordinates of B and the coordinates of C.

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

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 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 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