Worksheet: Finding the Equation of a Circle 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=10 and 𝑦=6 are two tangents to the circle.

  • A(𝑥18)+(𝑦6)=64
  • B(𝑥10)+(𝑦6)=64
  • C(𝑥10)+(𝑦14)=64
  • D(𝑥18)+(𝑦14)=64

Q2:

If the 𝑥-axis is a tangent to a circle whose equation is 𝑥+𝑦+𝑚𝑥+18𝑦124𝑚=0, find all possible values of 𝑚.

  • A𝑚=18 or 𝑚=4
  • B𝑚=36 or 𝑚=4
  • C𝑚=4
  • D𝑚=12 or 𝑚=4

Q3:

Find the general form of the equation of a circle that is tangent to the 𝑥-axis and has center (19,2).

  • A𝑥+𝑦+19𝑥2𝑦+361=0
  • B𝑥+𝑦+38𝑥4𝑦+361=0
  • C𝑥+𝑦+38𝑥4𝑦+369=0
  • D𝑥+𝑦2𝑥+19𝑦+361=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?

  • A𝑥+𝑦5𝑥+5𝑦5=0
  • B𝑥+𝑦5𝑥+5𝑦+25=0
  • C𝑥+𝑦10𝑥+10𝑦+75=0
  • D𝑥+𝑦10𝑥+10𝑦+25=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?

  • A𝑥+𝑦6𝑥+6𝑦+9=0
  • B𝑥+𝑦+3𝑥+3𝑦+9=0
  • C𝑥+𝑦+6𝑥+6𝑦+9=0
  • D𝑥+𝑦6𝑥6𝑦+9=0

Q6:

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

  • A(𝑥+5)+(𝑦5)=25, (𝑥+29)+(𝑦29)=841
  • B(𝑥5)+(𝑦+5)=25, (𝑥29)+(𝑦+29)=841
  • C(𝑥5)+(𝑦+5)=5, (𝑥29)+(𝑦+29)=29
  • D(𝑥+5)+(𝑦5)=10, (𝑥+29)+(𝑦29)=58

Q7:

Determine the equation of a circle with center (6,1) and tangent to the line 3𝑥+𝑦+21=0.

  • A(𝑥6)+(𝑦1)=160
  • B(𝑥+6)+(𝑦+1)=160
  • C(𝑥6)+(𝑦1)=10
  • D(𝑥+6)+(𝑦+1)=40

Q8:

Circle 𝑀 in the figure is tangent to the axes at points 𝐴 and 𝐵, and also tangent to the line 12𝑥+5𝑦60=0 at 𝐶. Find the equation of the circle.

  • A(𝑥+2)+(𝑦2)=100
  • B(𝑥2)+(𝑦2)=4
  • C(𝑥2)+(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.

  • A𝑥+𝑦𝑥4𝑦8=0
  • B𝑥+𝑦+2𝑥+8𝑦8=0
  • C𝑥+𝑦2𝑥8𝑦+42=0
  • D𝑥+𝑦2𝑥8𝑦8=0

Q10:

The 𝑦-axis is tangent to the circle 𝑥+𝑦+12𝑥+𝑚𝑦+49=0. What are the possible values of 𝑚?

  • A7
  • B12
  • C14,14
  • D49,49

Q11:

Determine the general form of the equation of circle 𝑀, given that it touches the two lines 𝑦=4 and 𝑦=10 and that the straight line 𝐿𝑥𝑦=7 passes through its center.

  • A𝑥+𝑦+8𝑥+6𝑦24=0
  • B𝑥+𝑦+4𝑥+3𝑦24=0
  • C𝑥+𝑦+28𝑥14𝑦+196=0
  • D𝑥+𝑦8𝑥6𝑦24=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.

  • A𝑥+𝑦+36𝑥18𝑦+18=0
  • B𝑥+𝑦36𝑥+18𝑦+324=0
  • C𝑥+𝑦+18𝑥36𝑦+81=0
  • D𝑥+𝑦36𝑥+18𝑦+18=0

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).

  • A𝑥+𝑦8𝑥+16𝑦+64=0
  • B𝑥+𝑦8𝑥16𝑦120=0
  • C𝑥+𝑦12𝑥12𝑦184=0
  • D𝑥+𝑦8𝑥+16𝑦48=0

Q14:

Are the two circles 𝐶𝑥+𝑦16𝑥+2𝑦+40=0 and 𝐶𝑥+𝑦+8𝑥+2𝑦+1=0 externally tangent?

  • Ayes
  • Bno

Q15:

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

  • A2 length units
  • B10 length units
  • C7 length units
  • D1 length unit

Q16:

Circles (𝑥14)+(𝑦+4)=𝑘 and (𝑥+5)+(𝑦+4)=25 have a common tangent. What are the possible values of 𝑘?

  • A28 or 48
  • B196 or 576
  • C14 or 24
  • D25 or 625

Q17:

The straight line 𝐿 touches the circle 𝑀 at 𝐴(2,8) and intersects the 𝑥-axis at 𝐵3415,0. Determine the equation of the circle 𝑀.

  • A(𝑥17)+(𝑦17)=289
  • B(𝑥17)+𝑦=17
  • C(𝑥+17)+𝑦=289
  • D(𝑥+17)+𝑦=17

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