Closest point on a graph Problem set A

Rational functions have asymptotes.

1. (a) Write down the equations of the vertical and horizontal asymptotes of the graph of \(f(x)=\dfrac{9x}{x-6}\), where \(x\neq6\).
2. (b) The graph of \(g(x)=\dfrac{8x}{x-q}\) has asymptotes that intersect at the point \(Q(-2,\,8)\). Find the value of q.

The graph of \(f(x)=\dfrac{4x}{x-3}\) has asymptotes that meet at \(Q(3,\,4)\). A point \(P(x,\,y)\) on the graph is a distance \(PQ=\sqrt{(x-3)^2+\left(\dfrac{12}{x-3}\right)^2}\) 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 drone flies along a path modelled by \(y=\sqrt{x}\), where x and y are in kilometres. A radio mast stands at the point M(9, 0).

1. (a) Write an expression for the distance \(d\) between the point \((x,\,\sqrt{x})\) on the curve and the mast M(9, 0).
2. (b) Find the shortest distance from the mast M(9, 0) to the curve.

A bird colony on an island has \(P(t)=\dfrac{600t}{t+4}\) birds, where t is the number of years since monitoring began, \(t\ge0\).

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

The point \(P(x,\,y)\) lies on the curve \(y=x^2\). The point A is \((0,\,5)\). Find the coordinates of the points on the curve that are closest to A.