Closest point on a graph Problem set B

Rational functions have asymptotes.

1. (a) Write down the equations of the vertical and horizontal asymptotes of the graph of \(f(x)=\dfrac{4x}{x+3}\), where \(x\neq-3\).
2. (b) The graph of \(g(x)=\dfrac{6x}{x-q}\) has asymptotes that intersect at the point \(Q(5,\,6)\). Find the value of q.

The point \(P(x,\,y)\) lies on the graph of \(f(x)=\dfrac{3x}{x+2}\). The asymptotes of the graph meet at \(Q(-2,\,3)\). Show that \(PQ=\sqrt{(x+2)^2+\left(\dfrac{-6}{x+2}\right)^2}\).

The graph of \(f(x)=\dfrac{3x}{x+2}\) has asymptotes that meet at \(Q(-2,\,3)\). A point \(P(x,\,y)\) on the graph is a distance \(PQ=\sqrt{(x+2)^2+\left(\dfrac{-6}{x+2}\right)^2}\) from Q.

1. (a) Use your calculator to find the x-coordinates of the two points on the graph closest to Q.
2. (b) Find the coordinates of these points and the shortest distance PQ.

A fence runs along the curve \(y=\sqrt{x}\), where x and y are in metres. A sensor is fixed at the point S(11, 0).

1. (a) Write an expression for the distance \(d\) between the point \((x,\,\sqrt{x})\) on the curve and the sensor S(11, 0).
2. (b) Find the shortest distance from the sensor S(11, 0) to the curve.

The concentration of a drug in the blood is modelled by \(C(t)=\dfrac{12t}{t+3}\) mg per litre, where t is the number of hours after the dose, \(t\ge0\).

1. (a) Write down the equations of the vertical and horizontal asymptotes of the graph of C.
2. (b) What does the horizontal asymptote tell you about the concentration?

The point \(P(x,\,y)\) lies on the curve \(y=\dfrac{32}{x}\), where \(x>0\). O is the origin.

1. (a) Show that \(OP=\sqrt{x^2+\dfrac{1024}{x^2}}\).
2. (b) Find the coordinates of the point on the curve closest to O, and the distance OP.

The point \(P(x,\,y)\) lies on the curve \(y=x^2\). The point A is \((0,\,3)\). Find the coordinates of the points on the curve that are closest to A.