Tangents & normals

In this question you must show detailed reasoning.

A curve has parametric equations x = t³ + t², y = t² + 2t for all real values of t.

The curve passes through the point P with coordinates (2, 3).

Show that the equation of the tangent to the curve at the point P can be written as 5 y = 4 x + 7

Determine the coordinates of the point where the tangent to the curve at P meets the curve

again.

The graph of y = e^x can be transformed to the graph of y = e^2x-1 by a stretch parallel to the x-axis followed by a translation.

State the scale factor of the stretch.

Give full details of the translation.

Alternatively the graph of y = e^x can be transformed to the graph of y = e^2x-1 by a stretch parallel to the x-axis and a stretch parallel to the y-axis.

State the scale factor of the stretch parallel to the y-axis.

The point P lies on the curve y = e^2x-1 and has x-coordinate of

Show that the tangent to the curve y = e^2x-1 at P has equation y = 2x.

Find the exact area enclosed by the curve y = e^2x-1, the tangent to the curve at P and the

y-axis.

Express 3x² - 12x + 17 in the form a(x - b)² + c where a, b and c are constants.

State the coordinates of the minimum point of the curve y = 3x² - 12x + 17.

State the equation of the normal to the curve y = 3x² - 12x + 17 at its minimum point.

The diagram shows part of the graph of y = x². The normal to the curve at the point A(1, 1) meets the curve again at B. Angle AOB is denoted by α.

Determine the coordinates of B.

Hence determine the exact value of tan α.

Show that the equation of the tangent at P can be written as y =-2p²x + 4p.

The tangent to this curve at P crosses the x-axis at the point A and the normal to this curve at P crosses the x-axis at the point B.

Show that the ratio PA: PB is 1:2p².

In this question you must show detailed reasoning.

The diagram shows the curve with equation 4xy = 2(x² + 4y²) - 9x.

At the point P on the curve the tangent to the curve is parallel to the y-axis and at the point Q on the curve the tangent to the curve is parallel to the x-axis.

A curve has equation x³ - 3x²y + y² + 1 = 0.

Find the equation of the normal to the curve at the point (1, 2).

By considering show that the curve is convex for all values of x.