Higher Maths: Circles in Contact and the Derived Graph

The circle is one of the problem-solving topics of Higher Mathematics, and its hardest questions are about contact: a line that may or may not be a tangent, and two circles that may touch or cross. This material covers the circle skills listed in the Higher Mathematics course specification under applying algebraic skills to circles, using properties of tangency and determining the intersection of circles or a line and a circle, together with sketching the graph of $y=f^{\mathrm{\prime }}(x)$ given the graph of $y=f(x)$ from the functions skills.

The first part works with a line and a circle. You substitute a line into a circle's equation to find the points where they meet, use the discriminant of the resulting quadratic to decide whether the line cuts, touches or misses the circle, find the values of a constant that make a line a tangent, and identify the point of contact from the repeated root. You also find the equation of the tangent at a given point on a circle from the perpendicular radius.

The second part compares two circles. You find each centre and radius and compare the distance between the centres with the sum and the difference of the radii to decide whether the circles touch externally, touch internally or intersect. You then find the point of contact of two touching circles on the line of centres, and the two intersection points of crossing circles by subtracting their equations to obtain the common chord. The category's straight line and circle material covers the basic equation of a circle and simple tangency; this one builds on it.

The last part is about the derived graph. Because every graph is described in words and coordinates rather than drawn, you translate a description of $y=f(x)$ into the key features of $y=f^{\mathrm{\prime }}(x)$: its roots at the stationary points, its sign on each interval, its degree, and how a maximum, a minimum or a point of inflexion shows up as a crossing or a touch. You also reason the other way, from the derived graph to the nature of a stationary point.

The material offers three formats. The quiz has 12 questions with worked explanations that check each answer by substitution. The flashcards collect the tangency test, the rules for touching circles and the link between $f$ and $f^{\mathrm{\prime }}$. The printable written work has 8 longer tasks with exact answers, including intersections with surds, a pair of touching circles and a full description of a derived graph.

It is aimed at S5 and S6 students preparing for Higher Mathematics.

• Find the points of intersection of a line and a circle
• Use the discriminant to decide whether a line is a tangent, and find unknowns that make it one
• Find the equation of a tangent at a point on a circle
• Decide whether two circles touch externally, touch internally or intersect
• Find the point of contact of touching circles and the intersection points of two circles
• Sketch or describe y = f′(x) from the features of y = f(x), and reason back from f′ to f

Practice material written by Zestly, based on the Qualifications Scotland (formerly SQA) Higher Mathematics course specification, version 3.0 (algebraic and geometric skills: applying algebraic skills to circles — properties of tangency, intersection of circles or a line and a circle; algebraic skills: sketching y = f′(x) given the graph of y = f(x)).