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Intersection of straight lines and circles

Intersection of straight lines and circles

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Tutor: Daniel

Summary

Intersection of straight lines and circles

In a nutshell

Algebra is used to find the coordinates of the point of intersection of a circle and a straight line. Straight lines can intersect a circle once, twice, or not at all. A tangent is a line that touches a circle at one point.



Finding points of intersection 

To find the points of intersection, the equations of the circle and the straight line can be solved simultaneously. 


Example 1

Find the coordinates of the points where the line y=x+4y=x+4y=x+4​ intersects the circle (x−3)2+(y−5)2 = 34\left(x-3\right)^{2}+\left(y-5\right)^{2}\ =\ 34(x−3)2+(y−5)2 = 34.


First, substitute the line equation into the quadratic to solve for xxx or yyy.

(x−3)2+((x+4)−5)2= 34(x−3)2+(x−1)2=34\begin{aligned}\left(x-3\right)^{2}+\left((x+4)-5\right)^{2} &=\ 34 \\\left(x-3\right)^{2}+\left(x-1\right)^{2} &= 34\end{aligned}(x−3)2+((x+4)−5)2(x−3)2+(x−1)2​= 34=34​


Expand the brackets to collect terms.

(x2−6x+9)+(x2−2x+1)=342x2−8x+10=342x2−8x−24=0x2−4x−12=0(x−6)(x+2)=0\begin{aligned}({x^2} - 6x+9) + (x^2 -2x +1)&=34 \\2x^2-8x+10&=34 \\2x^2-8x-24 &=0 \\x^2 - 4x-12&=0\\(x-6)(x+2)&=0\end{aligned}(x2−6x+9)+(x2−2x+1)2x2−8x+102x2−8x−24x2−4x−12(x−6)(x+2)​=34=34=0=0=0​

​​​​

Solve for xxx.​

(x−6)(x+2)=0(x-6)(x+2)=0(x−6)(x+2)=0​​

x=6x=6x=6​ and x=−2x=-2x=−2

Substitute the xxx terms into the line equation to solve for yyy​. ​

y=6+4=10y=6+4=10y=6+4=10

y=−2+4=2y = -2+4 =2y=−2+4=2

Therefore, the line y=x+4y=x+4y=x+4 intersects the circle at points​

(6,10)‾\underline{(6,10)}(6,10)​ and (−2,2)‾\underline{(-2,2)}(−2,2)​

Maths; Circles; KS5 Year 12; Intersection of straight lines and circles



Finding the number of intersection points

Solve the line and circle equations simultaneously to find the discriminant b2−4acb^2-4ac b2−4ac to test for roots of the quadratic equation. 


​b2−4ac>0b^2-4ac >0b2−4ac>0​​
There are two distinct roots, therefore, two points of intersection. 
​b2−4ac=0b^2-4ac =0b2−4ac=0​​
There is a repeated root, therefore, one point of intersection. 
​b2−4ac<0b^2-4ac <0b2−4ac<0​​
There are no real roots, therefore, no points of intersection.

Maths; Circles; KS5 Year 12; Intersection of straight lines and circles

​

​Example 2

Show that the line y=x−7y= x-7y=x−7 does not meet the circle (x+3)2+y2=42(x+3)^2+y^2=42(x+3)2+y2=42​.


First, substitute line y=x−7y= x-7y=x−7 into the equation of the circle and collect terms.  ​

(x+3)2+(x−7)2=42x2+6x+9+x2−14x+49=422x2−8x+16=0x2−4x+8=0\begin{aligned}(x+3)^2+(x-7)^2&=42 \\x^2+6x+9+x^2-14x+49&=42 \\2x^2-8x+16&=0 \\x^2-4x+8 &= 0\end{aligned}(x+3)2+(x−7)2x2+6x+9+x2−14x+492x2−8x+16x2−4x+8​=42=42=0=0​​​

​​

Use the discriminant of the quadratic to test for roots. 

b2−4ac=(−4)2−(4×1×8)=−16b^2-4ac = (-4)^2-(4\times1\times8) = -16 b2−4ac=(−4)2−(4×1×8)=−16

 −16<0-16<0−16<0, therefore, there are no points of intersection. 

​

Maths; Circles; KS5 Year 12; Intersection of straight lines and circles
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Equation of a straight line: y = mx + c

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Equation of a straight line: y = mx + c

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Equation of a straight line: y = mx + c

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Intersection of straight lines and circles

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FAQs - Frequently Asked Questions

What is a tangent?

A tangent is a line that touches a circle at a point.

How many times can a straight line intersect a circle?

Straight lines can intersect a circle once, twice, or not at all.

How can the coordinates of the point of intersection of a circle and a straight line be found?

Algebra is used to find the coordinates of the point of intersection of a circle and a straight line.

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