Worksheet: Equations of Circles Using Tangents

In this worksheet, we will practice finding the equation of a circle using tangents.

Q1:

Determine the equation of a circle whose radius equals 8 length units and center 𝑀 lies in the first quadrant, given that the two straight lines 𝑥 = 1 0 and 𝑦 = 6 are two tangents to the circle.

  • A ( 𝑥 1 8 ) + ( 𝑦 1 4 ) = 6 4
  • B ( 𝑥 1 8 ) + ( 𝑦 6 ) = 6 4
  • C ( 𝑥 1 0 ) + ( 𝑦 6 ) = 6 4
  • D ( 𝑥 1 0 ) + ( 𝑦 1 4 ) = 6 4

Q2:

If the 𝑥 -axis is a tangent to a circle whose equation is 𝑥 + 𝑦 + 𝑚 𝑥 + 1 8 𝑦 1 2 4 𝑚 = 0 , find all possible values of 𝑚 .

  • A 𝑚 = 1 2 or 𝑚 = 4
  • B 𝑚 = 4
  • C 𝑚 = 3 6 or 𝑚 = 4
  • D 𝑚 = 1 8 or 𝑚 = 4

Q3:

Find the general form of the equation of a circle that is tangent to the 𝑥 -axis and has center ( 1 9 , 2 ) .

  • A 𝑥 + 𝑦 + 1 9 𝑥 2 𝑦 + 3 6 1 = 0
  • B 𝑥 + 𝑦 2 𝑥 + 1 9 𝑦 + 3 6 1 = 0
  • C 𝑥 + 𝑦 + 3 8 𝑥 4 𝑦 + 3 6 1 = 0
  • D 𝑥 + 𝑦 + 3 8 𝑥 4 𝑦 + 3 6 9 = 0

Q4:

Pulley 𝐵 is attached by a wire cable to pulley 𝐴 , which is tangent to both coordinate axes and has radius 5. What is the equation of this circle?

  • A 𝑥 + 𝑦 5 𝑥 + 5 𝑦 + 2 5 = 0
  • B 𝑥 + 𝑦 5 𝑥 + 5 𝑦 5 = 0
  • C 𝑥 + 𝑦 1 0 𝑥 + 1 0 𝑦 + 7 5 = 0
  • D 𝑥 + 𝑦 1 0 𝑥 + 1 0 𝑦 + 2 5 = 0

Q5:

A circle is in the third quadrant and touches the 𝑥 -axis at ( 3 , 0 ) . If it is also tangent to the 𝑦 -axis, what is the equation of the circle?

  • A 𝑥 + 𝑦 + 3 𝑥 + 3 𝑦 + 9 = 0
  • B 𝑥 + 𝑦 + 6 𝑥 + 6 𝑦 + 9 = 0
  • C 𝑥 + 𝑦 6 𝑥 6 𝑦 + 9 = 0
  • D 𝑥 + 𝑦 6 𝑥 + 6 𝑦 + 9 = 0

Q6:

Give the equations of all circles tangent to the coordinate axes and passing through ( 8 , 9 ) .

  • A ( 𝑥 5 ) + ( 𝑦 + 5 ) = 2 5 , ( 𝑥 2 9 ) + ( 𝑦 + 2 9 ) = 8 4 1
  • B ( 𝑥 + 5 ) + ( 𝑦 5 ) = 1 0 , ( 𝑥 + 2 9 ) + ( 𝑦 2 9 ) = 5 8
  • C ( 𝑥 5 ) + ( 𝑦 + 5 ) = 5 , ( 𝑥 2 9 ) + ( 𝑦 + 2 9 ) = 2 9
  • D ( 𝑥 + 5 ) + ( 𝑦 5 ) = 2 5 , ( 𝑥 + 2 9 ) + ( 𝑦 2 9 ) = 8 4 1

Q7:

Determine the equation of a circle with center ( 6 , 1 ) and tangent to the line 3 𝑥 + 𝑦 + 2 1 = 0 .

  • A ( 𝑥 6 ) + ( 𝑦 1 ) = 1 0
  • B ( 𝑥 + 6 ) + ( 𝑦 + 1 ) = 1 6 0
  • C ( 𝑥 + 6 ) + ( 𝑦 + 1 ) = 4 0
  • D ( 𝑥 6 ) + ( 𝑦 1 ) = 1 6 0

Q8:

Circle 𝑀 in the figure is tangent to the axes at points 𝐴 and 𝐵 , and also tangent to the line 1 2 𝑥 + 5 𝑦 6 0 = 0 at 𝐶 . Find the equation of the circle.

  • A ( 𝑥 + 2 ) + ( 𝑦 2 ) = 1 0 0
  • B ( 𝑥 + 2 ) + ( 𝑦 + 2 ) = 4
  • C ( 𝑥 2 ) + ( 𝑦 + 2 ) = 9
  • D ( 𝑥 2 ) + ( 𝑦 2 ) = 4

Q9:

Find the general form of the equation of a circle that passes through the two points 𝐴 ( 2 , 0 ) and 𝐵 ( 4 , 8 ) , given that the tangents at 𝐴 and 𝐵 are parallel.

  • A 𝑥 + 𝑦 2 𝑥 8 𝑦 + 4 2 = 0
  • B 𝑥 + 𝑦 𝑥 4 𝑦 8 = 0
  • C 𝑥 + 𝑦 + 2 𝑥 + 8 𝑦 8 = 0
  • D 𝑥 + 𝑦 2 𝑥 8 𝑦 8 = 0

Q10:

The 𝑦 -axis is tangent to the circle 𝑥 + 𝑦 + 1 2 𝑥 + 𝑚 𝑦 + 4 9 = 0 . What are the possible values of 𝑚 ?

  • A12
  • B7
  • C 4 9 , 4 9
  • D 1 4 , 1 4

Q11:

Determine the general form of the equation of circle 𝑀 , given that it touches the two lines 𝑦 = 4 and 𝑦 = 1 0 and that the straight line 𝐿 𝑥 𝑦 = 7 passes through its center.

  • A 𝑥 + 𝑦 + 4 𝑥 + 3 𝑦 2 4 = 0
  • B 𝑥 + 𝑦 + 8 𝑥 + 6 𝑦 2 4 = 0
  • C 𝑥 + 𝑦 + 2 8 𝑥 1 4 𝑦 + 1 9 6 = 0
  • D 𝑥 + 𝑦 8 𝑥 6 𝑦 2 4 = 0

Q12:

Find the general equation of the circle 𝑀 , given that it touches the 𝑥 -axis at 𝐵 and the straight line 𝑥 = 9 at 𝐶 .

  • A 𝑥 + 𝑦 3 6 𝑥 + 1 8 𝑦 + 1 8 = 0 2 2
  • B 𝑥 + 𝑦 + 3 6 𝑥 1 8 𝑦 + 1 8 = 0 2 2
  • C 𝑥 + 𝑦 3 6 𝑥 + 1 8 𝑦 + 3 2 4 = 0 2 2
  • D 𝑥 + 𝑦 + 1 8 𝑥 3 6 𝑦 + 8 1 = 0 2 2

Q13:

Determine the equation of the circle having its center at point ( 4 , 8 ) and touching the straight line passing through the two points ( 4 , 8 ) and ( 6 , 6 ) .

  • A 𝑥 + 𝑦 8 𝑥 + 1 6 𝑦 4 8 = 0
  • B 𝑥 + 𝑦 1 2 𝑥 1 2 𝑦 1 8 4 = 0
  • C 𝑥 + 𝑦 8 𝑥 1 6 𝑦 1 2 0 = 0
  • D 𝑥 + 𝑦 8 𝑥 + 1 6 𝑦 + 6 4 = 0

Q14:

Are the two circles 𝐶 𝑥 + 𝑦 1 6 𝑥 + 2 𝑦 + 4 0 = 0 and 𝐶 𝑥 + 𝑦 + 8 𝑥 + 2 𝑦 + 1 = 0 externally tangent?

  • Ano
  • Byes

Q15:

Given that the two straight lines 𝑦 = 6 and 𝑦 = 8 are tangent to a circle, determine its radius.

Q16:

Circles ( 𝑥 1 4 ) + ( 𝑦 + 4 ) = 𝑘 and ( 𝑥 + 5 ) + ( 𝑦 + 4 ) = 2 5 have a common tangent. What are the possible values of 𝑘 ?

  • A28 or 48
  • B25 or 625
  • C14 or 24
  • D196 or 576

Q17:

The straight line 𝐿 touches the circle 𝑀 at 𝐴 ( 2 , 8 ) and intersects the 𝑥 -axis at 𝐵 3 4 1 5 , 0 . Determine the equation of the circle 𝑀 .

  • A ( 𝑥 1 7 ) + 𝑦 = 1 7
  • B ( 𝑥 1 7 ) + ( 𝑦 1 7 ) = 2 8 9
  • C ( 𝑥 + 1 7 ) + 𝑦 = 2 8 9
  • D ( 𝑥 + 1 7 ) + 𝑦 = 1 7

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