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Intersections and tangents of a line and a circle

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Which discriminant result shows that a line is tangent to a circle?

\[\Delta=b^2-4ac\]
Year 11, VCE Units 1 & 2, Chapter 6, 1 difficulty level, Calculator allowedBuild a worksheet

Line and circle contact

Substitution changes a line and circle problem into a quadratic equation. The discriminant counts the common points, while a tangent radius is perpendicular to the tangent line

\[\Delta=b^2-4ac,\qquad m_rm_t=-1\text{ when both gradients exist}\]
  1. Substitute the line Replace the line variable in the circle equation to obtain a quadratic \(\displaystyle \left(x-1\right)^2+\left(y+2\right)^2=25,\ y=2x+1\Rightarrow\left(x-1\right)^2+\left(2x+3\right)^2=25\)
  2. Read the discriminant A negative, zero or positive discriminant gives zero, one or two common points respectively \(\displaystyle \Delta<0\Rightarrow0,\quad\Delta=0\Rightarrow1,\quad\Delta>0\Rightarrow2\)
  3. Use perpendicular gradients When both gradients exist, the radius and tangent gradients are negative reciprocals \(\displaystyle m_r=2\Rightarrow m_t=-\frac{1}{m_r}=-\frac{1}{2}\)
  4. Subtract squared lengths In the right triangle, the tangent length is found after subtracting the squared radius from the squared distance to the external point \(\displaystyle CP=13,\ CT=5\Rightarrow PT^2=CP^2-CT^2=13^2-5^2=144\)

Worked example

The circle and line are given below. Determine how many common points they have

\[\left(x-2\right)^2+\left(y+1\right)^2=25,\quad y=3x-4\]
  1. \(\displaystyle \left(x-2\right)^2+\left(3x-3\right)^2=25\)Substitute the line equation into the circle equation
  2. \(\displaystyle 10x^2-22x-12=0\)Expand both squares and collect the terms in x.
  3. \(\displaystyle \Delta=(-22)^2-4(10)(-12)=964\)Use the quadratic coefficients in the discriminant formula
  4. \(\displaystyle \Delta=964>0\Rightarrow\text{two distinct common points}\)A positive discriminant gives two distinct real solutions

Common mistakes

Discriminant sign is matched to the wrong contact count
✕\(\displaystyle \Delta>0\Rightarrow\text{the line is tangent}\)✓\(\displaystyle \Delta=0\Rightarrow\text{the line is tangent}\)
A tangent has one repeated intersection, so the discriminant is zero
Tangent gradient is set equal to radius gradient
✕\(\displaystyle m_r=2\Rightarrow m_t=2\)✓\(\displaystyle m_r=2\Rightarrow m_t=-\frac{1}{2}\)
The radius and tangent are perpendicular, so their gradients are negative reciprocals
Squared lengths are added for a tangent
✕\(\displaystyle PT=\sqrt{13^2+5^2}=\sqrt{194}\)✓\(\displaystyle PT=\sqrt{13^2-5^2}=\sqrt{144}=12\)
By Pythagoras, the squared centre-to-external-point distance is the sum of the squared radius and squared tangent length, so subtract the squared radius to find the squared tangent length

Difficulty levels

challengeUse exact substitution, discriminants, perpendicular gradients or tangent right triangles to analyse line-circle contact

Questions about this topic

What does the discriminant say about a line and a circle?+

After substitution gives a quadratic, a negative discriminant means no common points, zero means one repeated point, and a positive discriminant means two common points

Why is a tangent perpendicular to the radius?+

The shortest distance from the centre to a line is perpendicular to the line, and the tangent touches the circle at the endpoint of that radius

Why is the tangent length found by subtracting squares?+

The centre, contact point and external point form a right triangle, so subtract the squared radius from the squared centre-to-external-point distance to find the squared tangent length

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 6: Functions and relations

In this chapter

Classify a relation and state its domain and rangeEvaluate and determine function rulesFind the maximal domain of an algebraic functionAnalyse reciprocal and square-reciprocal functionsAnalyse sideways-parabola and root-function familiesDetermine a power-family rule from features and pointsDetermine a circle's equation and featuresAnalyse semicircles and circular regionsApply and describe transformations of a functionIntersections and tangents of a line and a circleEvaluate and interpret a restricted rational modelForm and analyse an inverse relation or function