We have now sorted the straight line equations from the Warm-up section into pairs of lines which are perpendicular.

\(2y-x-3=0\) and \(y+2x+1=0\), intersecting at \((-1,1)\)

\(y-3x-2=0\) and \(3y+x-6=0\), intersecting at \((0,2)\)

\(y-8x+6=0\) and \(16y+2x+31=0\), intersecting at \(\bigl(\frac{1}{2},-2 \bigr)\)

\(y+x+2=0\) and \(y-x-2=0\), intersecting at \((-2,0)\)

Each of these pairs of lines are the tangent and normal lines to a different curve at a point. Below are the gradient functions (derivatives) for each of the original curves.

\(2x+5\)

\(15x^2+12x+3\)

\(\dfrac{-2}{x^2}\)

\(24x^2-8x+6\)

Can you determine which of these gradient functions (derivatives) corresponds to each pair of tangent and normal lines given above?

For each pair, can you determine which of the straight lines is the tangent and which is the normal?