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Lesson Worksheet: Equation of an Ellipse Mathematics • 10th Grade

In this worksheet, we will practice finding the equation of an ellipse using different givens and using it to solve problems that involve ellipse-shaped constructions.


Derive the equation of an ellipse with foci at the points (1,3) and (6,3), which has a major axis of length 15.

  • A(2π‘₯βˆ’3)225+(π‘¦βˆ’7)50=1
  • B(2π‘₯βˆ’7)225+(π‘¦βˆ’3)50=1
  • C(2π‘₯βˆ’7)50+(π‘¦βˆ’3)225=1
  • D(π‘₯βˆ’7)225+(π‘¦βˆ’3)50=1
  • E(2π‘₯βˆ’3)50+(π‘¦βˆ’7)225=1


An ellipse has a vertex at (11,5) and a covertex at (6,8). The major axis of the ellipse is parallel to the π‘₯-axis. What is the equation of the ellipse?

  • Aπ‘₯25+𝑦9=1
  • B(π‘₯βˆ’6)25+(π‘¦βˆ’5)9=1
  • C(π‘₯βˆ’6)5+(π‘¦βˆ’5)3=1
  • Dπ‘₯9+𝑦25=1
  • E(π‘₯βˆ’5)25+(π‘¦βˆ’6)9=1


Derive the equation of an ellipse centered at the origin with foci at the points (4,0) and (βˆ’4,0), which has a major axis of length 10.

  • Aπ‘₯5+𝑦3=1
  • Bπ‘₯25+𝑦9=1
  • Cπ‘₯9+𝑦25=1
  • Dπ‘₯16+𝑦25=1
  • Eπ‘₯25+𝑦16=1


Derive the equation of an ellipse centered at the origin with foci at the points (2,0) and (βˆ’2,0), which has its vertices at the points (8,0) and (βˆ’8,0).

  • Aπ‘₯16+𝑦64=1
  • Bπ‘₯48+𝑦16=1
  • Cπ‘₯25+𝑦64=1
  • Dπ‘₯64+𝑦60=1
  • Eπ‘₯48+𝑦64=1


Fill in the blank: The ellipse 9π‘₯+16𝑦=1 has a major axis of length cm.

  • A13
  • B23
  • C6
  • D12
  • E8


Given that the simultaneous equations (π‘₯βˆ’2)5+(π‘¦βˆ’6)=16,𝑦=49π‘₯+829 are graphically represented in the figure, solve the two equations and find the most accurate estimations of π‘₯- and 𝑦-values.

  • Aπ‘₯=βˆ’7 and 𝑦=6, π‘₯=2 and 𝑦=10
  • Bπ‘₯=2 and 𝑦=βˆ’16, π‘₯=14 and 𝑦=11
  • Cπ‘₯=6 and 𝑦=βˆ’7, π‘₯=10 and 𝑦=2
  • Dπ‘₯=βˆ’16 and 𝑦=2, π‘₯=11 and 𝑦=14
  • Eπ‘₯=2 and 𝑦=βˆ’7, π‘₯=0 and 𝑦=9


What is the equation of an ellipse that has a semimajor axis, which is parallel to the 𝑦-axis, of length 10 and foci points (1,5) and (1,βˆ’3)?

  • Aπ‘₯84+𝑦100=1
  • B(π‘₯+1)84+(𝑦+1)100=1
  • C(π‘₯+1)100+(𝑦+1)84=1
  • D(π‘₯βˆ’1)100+(π‘¦βˆ’1)84=1
  • E(π‘₯βˆ’1)84+(π‘¦βˆ’1)100=1


Let (βˆ’2,βˆ’1), (βˆ’6,βˆ’1), and (βˆ’2,βˆ’4) be the coordinates of the center, a focus, and a covertex of an ellipse respectively. What is the equation describing this ellipse?

  • A(π‘₯βˆ’2)9+(𝑦+1)25=1
  • B(π‘₯+1)25+(𝑦+2)9=1
  • C(π‘₯+2)25+(𝑦+1)9=1
  • D(π‘₯+1)9+(𝑦+2)25=1
  • E(π‘₯βˆ’2)25+(π‘¦βˆ’1)9=1


An ellipse has the equation 8π‘₯+196𝑦=1,568. If the major axis is reduced to half, the minor axis is doubled, and the center does not change, what will the equation of the new ellipse be?

  • Aπ‘₯32+𝑦49=1
  • Bπ‘₯7+𝑦32=1
  • Cπ‘₯49+𝑦16=1
  • Dπ‘₯32+𝑦7=1
  • Eπ‘₯49+𝑦32=1


Let 𝐴 be an ellipse whose equation is given by 20(π‘₯+1)+36(π‘¦βˆ’1)=720. If 𝐡 is another ellipse with its center at the focus of 𝐴 that is closer to the origin and has the same length of major and minor axes as 𝐴, derive the equation of 𝐡.

  • A20(π‘₯βˆ’3)+36(π‘¦βˆ’1)=720
  • B36(π‘₯βˆ’3)+20(π‘¦βˆ’1)=720
  • C20(π‘₯βˆ’3)+36(𝑦+1)=720
  • D20(π‘₯+5)+36(𝑦+1)=720
  • E36(π‘₯βˆ’5)+20(π‘¦βˆ’1)=720

This lesson includes 1 additional question and 54 additional question variations for subscribers.

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