Harder tangent and normal problems Problem set A

Use technology to answer each part. Give each answer in the form \(y=mx+c\), with m and c to 3 significant figures.

1. (a) Find the equation of the tangent to \(y=x\sqrt{x+5}\) at \(x=4\).
2. (b) Find the equation of the normal to \(y=\frac{1}{x+6}\) at \(x=-4\).

Use technology to find the equation of the normal to each curve at the given point.

1. (a) \(y=x^2e^{3x}\) at \(x=0\)
2. (b) \(y=(x-2)^2e^{x}\) at \(x=2\)

Give each answer in the form \(y=mx+c\).

1. (a) Find the equation of the tangent to the curve \(y=x^2-3x\) which is parallel to the line \(y=5x+1\).
2. (b) Find the equation of the normal to the curve \(y=x^2-4x\) which is parallel to the line \(x+2y=6\).

Find the equations of the normals to the curve \(y=x^3-3x\) which are perpendicular to the line \(y=9x+1\). Give each in the form \(y=mx+c\).

Find the coordinates of the points on the curve \(y=x^3-3x^2+x\) at which the tangent is parallel to the line \(y=10x-4\).

Find all the points on the curve \(y=x^2-2x-2\) where the gradient of the tangent equals the y-coordinate.

Find the coordinates of the point where the normal to each curve at the given point meets the curve again.

1. (a) \(y=x^2+x\) at \(x=1\)
2. (b) \(y=\frac{2}{x}\) at \(x=1\)

A lamp is at the point \((3, -5)\). A ray of light leaves the lamp in a straight line that just touches the curve \(y=x^2+2\) (the line is a tangent to the curve).

1. (a) Find the coordinates of the two points where the line can touch the curve.
2. (b) Write down the equation of the tangent at the point with the larger x-coordinate, in the form \(y=mx+c\).

A curved cable is modelled by the curve \(y=\frac{a}{x}\), where a is a positive constant. The normal to the curve at the point where \(x=2\) is parallel to the line \(x-2y+3=0\). Find the value of a.

The tangent to the curve \(y=\frac{6}{x}\) at the point where \(x=a\) (\(a>0\)) meets the y-axis at P and the x-axis at Q. O is the origin. Show that the area of triangle OPQ does not depend on a.

Show that the tangent to the curve \(y=x^3-5x+1\) at the point with x-coordinate k meets the curve again at the point with x-coordinate \(-2k\).