Worksheet 7: Tangent Line of a Circle Name: Date: Tangent line of a circle can be determined once the tangent point or the slope of the line is known. Straight line: an overview General form : 0Ax By C+ + =

Slope-intercept form: , where slope y mx c= + 2 1

2 1

y y ymx x x

− Δ= =

− Δ, 1 1( , )x y and

2 2( , )x y are points on the line; c is the intercept. Example 1: Determine the equation of a line that passes through P(-2,5) and Q(4,-1)! Answer: General form:

( 2,5) 0 2 5 0P Ax By C A B C− → + + = → − + + =(4, 1) 0 4 0Q Ax By C A B C− → + + = → − + =

Eliminate C:

2 5 04 0

6 6 0

A B CA B C

A B A

− + + =− + = −

− + = → = B

03

Substitute: 2 5 0 2 5A B C A A C− + + = → − + + = 3 0A C C→ + = → = − A

3

Let: 1 1,A B C= → = = − So, the line is: 3 0x y+ − =

Slope - intercept form

5 ( 1) 6 12 4 6

ymx

Δ − −= = = = −

Δ − − −

Slope = -1 Substitute m and one point : ( 2,5)P y mx c− → = + 5 ( 1)( 2) 3c c→ = − − + → = Intercept = 3 Hence, the line is: 3y x= − + Is it the same result as in the general form?

Example 2: What is the slope of 2x + 3y – 6 = 0 ? Answer: Re-arrange 2 3 into slope-intercept form: 6x y+ − = 0

{{

6 2 2 22 3 6 0 23 3 c

m

xx y y y x y x−+ − = ⇔ = ⇔ = − ⇔ = − + 2

3

Hence, the slope is 23

− .

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Exercise

1. Determine the equation of a line that passes through the following points! a. P(1,2) and Q(1,-2)

f. A(0,2) and B(-1,-2)

b. M(1,3) and N(2,-1)

g. B(-3,-4) and C(3,4)

c. Q(-1,5) and R(2,7)

h. C(2,-3) and D(-2,3)

d. S (-2,-3) and T(-1,-1)

i. D(-2,-2) and E(-3,-3)

e. U(10,0) and V(0,10)

j. E(1,-5) and F(-5,1)

2. Determine the slope and intercept of lines in question 1!

Slope Intercept a

b

c

d

e

f

g

h

i

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j

3. Determine the lines with slope m and passes through the point Z. 1, (2,5)m Z= =

Does this line pass through (10,13)?

2, (0, 2)m Z= − = −

Does this line pass through (1,1)?

1 , (3,63

m Z= = )

Does this line pass through (12,9)?

2 , (3,55

m Z−= = )

Does this line pass through (-3,-5)?

5 , ( 12, 36)6

m Z= = − −

Does this line pass through (3,9)?

4. Rewrite the following line equations into slope-intercept form! a. 3 7 5x y+ − = 0

b. 2 12 03

x y+ − =

c. 7 5x y− + = 0

d. 1 102

x y− =

5. Rewrite the following line equations into general form!

a. 1 205

x y= +

b. 3 15y 4+ =

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c. 3 1y x− = 4

d. 15 12

0x y− =

Lines and Slopes Let:

Line 1 is g1 with slope of m1 Line 2 is g2 with slope of m2

Then: , i.e. g 1 and g2are parallel if and only if m1 = m2

, i.e. g 1 and g2 are perpendicular if and only if

g1 ⊥ g2

g1 // g2

m1×m2=-1

Example1: Find the equation of a line that is parallel to x + 3y = 2 and passes through (-1,0) Answer:

Line 1:

2 13 23 3

xx y y y x−+ = → = ⇔ = − +

23

→ slope = 113

m − . =

Line 2: parallel to line 1 → 2 113

m m= = −

Equation of line 2: slope 2

13

m = − and passes through (-1,0) is

( )1 10 1 1 3 1 03 3

y x yo x⇒ = − − + + =13 3

y mx c c c r⎛ ⎞= + → = − − + → = −⎟⎠

⎜⎝

Example 2: Find the equation of a line that is perpendicular to x + 3y = 2 and passes through (1,10). Answer:

Line 1: x + 3y = 2 → slope = 113

m = −

Line 2: perpendicular to line 1 → 1 2 2 211 13

m m m m 3= − ⇒ − = − ⇒ =

Hence; line 2 is 10 3 71y mx c c c 3 7y x= + → = × + ⇒ =→ = + Example 3: Prove that and 2 are perpendicular! 2 1x y+ = 100y x− + = 0

2

Proof: Line 1:

12 1 2 1x y y x m+ = → = − + → = −

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Line 2:

2100 1 12 100 0 502 2

xy x y y x m2

−− + = → = → = − → =

Since 1 2

12 12

m m = − × = − → 2 1x y+ = ⊥ 2 100y x 0− + =

EXERCISE Find the equation of lines that:

1. Parallel to and passes through (1,2)

2y x+ = 5 5

2. Perpendicular to 2y x+ = and passes through (1,2)

3. Parallel to and passes through (0,2)

3 7x y− + = 0 0

4. Perpendicular to 3 and passes through (0,2)

7x y− + =

5. Parallel to 1 2x y= + and passes through (2,3)

6. Perpendicular to 1 2x y= + and passes through (2,3)

Examine whether these lines are parallel or perpendicular!

1. 2 7 0 4 2 3x y and y x− + = − + = 0

2. 2 23 2 9 3 09

y x and y x= − − − + =

3. 2 5 17 0 5 2 13x y and x y+ + = = −

4. 1 1 4 0 3 2 3 02 3

x y and y x+ − = − + =

5. 3 7 0 2 43 0x y and y x− + = − − = 0

6. 5 9 0 4 3x y and y x− + = − + =

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Equation of a Circle’s Tangent Line Circle centered at (0,0) and radius r Circle centered at P(a,b) and radius r Let:

( , )Q QQ x y is ON the circle 2 2 2x y r+ = Then: Tangent line of the circle that passes through is given by: ( , )Q QQ x y

2Q Qx x y y r+ =

Let: ( , )Q QQ x y is ON the circle

( ) ( ) 222 rbyax =−+− Then: Tangent line of the circle that passes through

is given by: ( , )Q QQ x y

( )( ) ( )( ) 2Q Qx a x a y b y b r− − + − − =

Example 1: Find the equation of line that is tangent to the circle 2 2 25x y+ = at (3,-4). Answer: The tangent line is 2 3 4 2Q Qx x y y r x y+ = ⇔ − = 5 Example 2: Find the equation of line that is tangent to the circle ( ) ( ) 2021 22 =+++ yx at (3,-4). Answer: The tangent line is ( )( ) ( )( ) ( )( ) ( )( )2 3 1 1 4 2 2 20Q Qx a x a y b y b r x y− − + − − = ⇔ + + + − + + =

( ) ( )4 1 2 2 20 4 2 20 2 10x y x y x y⇔ + − + = ⇒ − = ⇒ − = Example 3: Find the equation of line that is tangent to the circle 2 22 2 4 8 3x y x y 0+ + + − = at

1 1,2 2

⎛ ⎞−⎜ ⎟⎝ ⎠

Answer: Re-formulate the circle into standard form:

2 2 2 22 2 4 8 3 0 2 4 2 8 3x y x y x x y y+ + + − = ⇔ + + + − = 0

( ) ( )

( ) ( )

2 2 2

2 2

3 32 4 0 22 2131 22

x x y y x x

x y

→ + + + − = ⇔ +

→ + + + =

2 4 01 4 1 4y y+ + −+ + − − =

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The tangent line is

( )( ) ( )( ) ( ) ( )2 1 11 1 2 22 2Q Qx a x a y b y b r x y⎛ ⎞ ⎛ ⎞− − + − − = ⇔ − + + + + + =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠132

( ) ( )1 5 131 2 5 11 13 52 2 2

x y x y x y⇔ + + + = ⇒ + + = ⇒ + = 2

EXERCISE

1. Determine the tangent line of a circle centered at (0,0) with radius r and passes through point M

a. 2, (1,3)r M=

b. 1, ( 1,3)r M= −

c. 2, (2, 1)r M= −

d. 2, ( 1, 2)r M= − −

2. Determine the tangent line of a circle centered at P(a,b) with radius r and

passes through point N a. (0,1), 2, (2, 1)P r N= −

b. (1, 2), 2 , ( 1, 2)P r N= − −

c. (1,1), 2, (1,3)P r N=

d. (2,3), 1, ( 1,3)P r N= −

3. Determine the tangent line of the following circle and passes through point Q a) , Q(-6,-8) 2 2 100x y+ =

b) 2 216 16 25x y+ = ,Q 31,4

⎛ ⎞−⎜ ⎟⎝ ⎠

c) ,Q(0,-1)2 2 6 1x y x+ − − = 0 0

d) 2 23 3 4 8 27x y x y+ − + − = , Q(1,2)

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Tangent Line of a Circle when the Slope is Known Circle centered at (0,0) and radius r Circle centered at P(a,b) and radius r Let Slope of circle 2 2 2x y r+ = is m Then The tangent line is given by 21y mx r m= ± +

Let Slope of circle ( ) ( ) 222 rbyax =−+− is m Then The tangent line is given by ( ) 21y b m x a r m− = − ± +

Example 1: Find the line with slope 4 that is tangent to the circle 2 2 25x y+ = Answer: The tangent line is:

4 5 17y x= ± ⇔ 4 5 17y x= + and 4 5 17y x= − Example 2: Find the tangent line of the circle ( ) ( )2 23 1x y 16− + + = that is parallel to y = 2x + 5 Answer: Line y = 2x + 5→ m = 2 So, the tangent line is:

( )1 2 3 4 5y x+ = − ± ⇔ 2 7 4 5y x= − + and 2 7 4 5y x= − − EXERCISE

1. Find the equation of the tangent line of a circle centered at (0,0) and radius r,

with slope m. a. 1, 1r m= =

b. 2, 2r m= = −

c. 12,2

r m= = −

d. 3, 4r m= =

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2. Find the equation of the tangent line of a circle centered at P(a,b) and radius r,

with slope m. a. (0, 2), 1, 1P r m= =

b. (1,1), 2, 2P r m= = −

c. 1( 1, 2), 2,2

P r m− = = −

d. ( 2,3), 3, 4P r m− = =

3. Find the equation of the tangent line of the following circles.

a. , // 2x + y = 10 2 216 16 25x y+ =

b. 2 216 16 25x y+ = , ⊥ 2x + y = 10

c. , // 3x-2y= 12 2 216 16 25x y+ =

d. 2 216 16 25x y+ = , ⊥3x -2y = 12

4. Find points on the circle ( ) ( )2 22 5x y 74+ + − = that are contained in the

tangent line with slope 5.

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