Lesson Worksheet: The Equation of a Tangent to a Circle Mathematics • 11th Grade

In this worksheet, we will practice finding the equation of a tangent to a circle using the coordinates of the circle’s center and a point of tangency.

Q1:

Suppose that 𝐴𝐡 is a diameter of a circle center (βˆ’7,4). If 𝐡(βˆ’8,6), what is the general equation of the tangent to the circle at 𝐴?

  • Aπ‘₯+2π‘¦βˆ’10=0
  • B2π‘₯βˆ’π‘¦βˆ’10=0
  • Cπ‘₯+2𝑦+10=0
  • Dπ‘₯βˆ’2𝑦+10=0

Q2:

A circle with center 𝑀 has a diameter 𝐢𝐷, where 𝐷 is on the circumference. Given that the coordinates of the points 𝑀 and 𝐷 are ο€Όβˆ’112,βˆ’1 and (βˆ’7,7), respectively, determine the equation of the tangent to the circle at the point 𝐢.

  • A𝑦=βˆ’163π‘₯βˆ’394
  • B𝑦=βˆ’163π‘₯βˆ’334
  • C𝑦=316π‘₯βˆ’394
  • D𝑦=316π‘₯βˆ’334

Q3:

The equation of the given graph is π‘₯+𝑦=100. Find the equation of the tangent at the point (6,βˆ’8).

  • A𝑦=18π‘₯βˆ’354
  • B𝑦=43π‘₯βˆ’16
  • C𝑦=βˆ’34π‘₯βˆ’72
  • D𝑦=34π‘₯βˆ’252
  • E𝑦=16π‘₯βˆ’9

Q4:

For the equation π‘₯+𝑦=25 find the equation of the tangent at the point (3,4).

  • A𝑦=13π‘₯+3
  • B𝑦=34π‘₯+74
  • C𝑦=43π‘₯
  • D𝑦=14π‘₯+134
  • E𝑦=βˆ’34π‘₯+254

Which of the following is the graph of the circle along with the tangent line found in the first part?

  • A
  • B
  • C
  • D
  • E

Q5:

A circle has the equation π‘₯+𝑦=25. Find the gradient of the tangent at the point (4,βˆ’3).

  • A43
  • Bβˆ’34
  • C14
  • D34
  • Eβˆ’13

Q6:

Suppose that 𝐴𝐡 is a diameter at a circle with center 𝑀, where 𝐴(7,4) and 𝐡(βˆ’1,βˆ’2). What is the general equation of the tangent to the circle at 𝐴?

  • A𝑦=34π‘₯βˆ’54
  • B𝑦=23π‘₯+133
  • C𝑦=βˆ’43π‘₯+403
  • D𝑦=βˆ’403π‘₯+43
  • E𝑦=43π‘₯βˆ’403

Q7:

A circle has the equation (π‘₯βˆ’4)+(𝑦+5)=5. Find the gradient of the tangent at the point 𝐴(2,βˆ’4).

  • A2
  • B23
  • Cβˆ’2
  • D12
  • Eβˆ’12

Q8:

A circle has the equation π‘₯+𝑦=26. If the equation of a tangent at point 𝑃 is found to be 𝑦=βˆ’5π‘₯+26, find 𝑃.

  • A(βˆ’1,βˆ’5)
  • B(βˆ’5,βˆ’1)
  • C(1,5)
  • D(5,1)
  • E(βˆ’5,1)

Q9:

Determine the equation of the tangent to the circle (π‘₯+4)+(π‘¦βˆ’5)=34 at the point 𝐷(1,2).

  • A𝑦=35π‘₯βˆ’75
  • B𝑦=βˆ’53π‘₯+113
  • C𝑦=βˆ’35π‘₯βˆ’75
  • D𝑦=53π‘₯+13
  • E𝑦=βˆ’37π‘₯+117

Q10:

Which of the following graphs represents the circle (π‘₯βˆ’4)+(π‘¦βˆ’3)=13 and the tangent with gradient βˆ’23?

  • A
  • B
  • C
  • D
  • E

This lesson includes 6 additional questions and 117 additional question variations for subscribers.

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