Worksheet: Equation of a Plane in Space

In this worksheet, we will practice finding the equation of a plane using the x-, y-, and z-intercepts.

Q1:

Determine the general equation of the plane that intersects the negative 𝑥 -axis at a distance of 2 from the origin, intersects the positive 𝑧 -axis at a distance of 3 from the origin, and passes through the point 𝐶 ( 9 , 4 , 4 ) .

  • A 1 1 𝑥 4 𝑦 4 𝑧 + 1 2 = 0
  • B 1 2 𝑥 4 1 𝑦 8 𝑧 2 4 = 0
  • C 2 𝑥 + 3 𝑧 6 = 0
  • D 1 2 𝑥 + 4 1 𝑦 8 𝑧 + 2 4 = 0
  • E 9 𝑥 4 𝑦 7 𝑧 + 1 8 = 0

Q2:

Find the general equation of the plane that is perpendicular to the plane 6 𝑥 + 3 𝑦 + 4 𝑧 + 4 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 5 , 0 , 0 ) and ( 0 , 1 , 0 ) respectively.

  • A 6 𝑥 + 3 𝑦 + 4 𝑧 + 3 0 = 0
  • B 4 𝑥 2 0 𝑦 9 𝑧 + 2 0 = 0
  • C 6 𝑥 + 3 𝑦 + 4 𝑧 3 = 0
  • D 4 𝑥 + 2 0 𝑦 9 𝑧 2 0 = 0
  • E 4 𝑥 + 3 𝑦 + 4 𝑧 3 = 0

Q3:

Find the general equation of the plane that passes through the point ( 8 , 9 , 9 ) and cuts off equal intercepts on the three coordinate axes.

  • A 𝑥 + 𝑦 + 𝑧 1 0 = 0
  • B 8 𝑥 9 𝑦 9 𝑧 = 0
  • C 𝑥 + 𝑦 + 𝑧 6 4 8 = 0
  • D 𝑥 + 𝑦 + 𝑧 + 1 0 = 0
  • E 8 𝑥 + 𝑦 + 𝑧 = 0

Q4:

Given that the plane 2 𝑥 + 6 𝑦 + 2 𝑧 = 1 8 intersects the coordinate axes 𝑥 , 𝑦 , and 𝑧 at the points 𝐴 , 𝐵 , and 𝐶 , respectively, find the area of 𝐴 𝐵 𝐶 .

  • A 3 1 5 2
  • B 2 7 1 1
  • C 3 1 9 2
  • D 2 7 1 1 2
  • E 2 1 9

Q5:

Find the equation of the plane cutting the coordinate axes at 𝐴 , 𝐵 , and 𝐶 , given that the intersection point of the medians of 𝐴 𝐵 𝐶 is ( 𝑙 , 𝑚 , 𝑛 ) .

  • A 𝑙 𝑥 + 𝑚 𝑦 + 𝑛 𝑧 = 3
  • B 𝑙 𝑥 + 𝑚 𝑦 + 𝑛 𝑧 = 1
  • C 𝑥 𝑙 + 𝑦 𝑚 + 𝑧 𝑛 = 1
  • D 𝑥 𝑙 + 𝑦 𝑚 + 𝑧 𝑛 = 3
  • E 𝑥 + 𝑦 + 𝑧 = 𝑙 + 𝑚 + 𝑛

Q6:

Find the equation of the plane whose 𝑥 -, 𝑦 -, and 𝑧 -intercepts are 7 , 3, and 4 , respectively.

  • A 𝑥 4 𝑦 7 𝑧 4 = 1
  • B 𝑥 3 𝑦 7 𝑧 4 = 1
  • C 𝑥 7 𝑦 4 + 𝑧 3 = 1
  • D 𝑥 7 + 𝑦 3 𝑧 4 = 1

Q7:

Determine the general equation of the plane that intersects the negative 𝑥 -axis at a distance of 5 from the origin, intersects the negative 𝑧 -axis at a distance of 6 from the origin, and passes through the point 𝐶 ( 6 , 1 , 2 ) .

  • A 𝑥 + 𝑦 2 𝑧 1 2 = 0
  • B 6 𝑥 1 6 𝑦 + 5 𝑧 3 0 = 0
  • C 5 𝑥 6 𝑧 + 1 8 = 0
  • D 6 𝑥 + 1 6 𝑦 + 5 𝑧 + 3 0 = 0
  • E 6 𝑥 + 𝑦 + 4 𝑧 3 0 = 0

Q8:

Determine the general equation of the plane that intersects the positive 𝑥 -axis at a distance of 6 from the origin, intersects the negative 𝑧 -axis at a distance of 2 from the origin, and passes through the point 𝐶 ( 4 , 4 , 3 ) .

  • A 2 𝑥 + 4 𝑦 3 𝑧 6 = 0
  • B 4 𝑥 + 7 𝑦 1 2 𝑧 + 2 4 = 0
  • C 3 𝑥 𝑧 + 9 = 0
  • D 4 𝑥 7 𝑦 1 2 𝑧 2 4 = 0
  • E 4 𝑥 + 4 𝑦 𝑧 2 4 = 0

Q9:

Determine the general equation of the plane that intersects the negative 𝑥 -axis at a distance of 6 from the origin, intersects the negative 𝑧 -axis at a distance of 2 from the origin, and passes through the point 𝐶 ( 6 , 6 , 9 ) .

  • A 4 𝑥 2 𝑦 3 𝑧 6 = 0
  • B 2 𝑥 5 𝑦 6 𝑧 + 1 2 = 0
  • C 3 𝑥 𝑧 2 7 = 0
  • D 2 𝑥 + 5 𝑦 6 𝑧 1 2 = 0
  • E 6 𝑥 6 𝑦 7 𝑧 + 3 6 = 0

Q10:

Determine the general equation of the plane that intersects the negative 𝑥 -axis at a distance of 4 from the origin, intersects the positive 𝑧 -axis at a distance of 7 from the origin, and passes through the point 𝐶 ( 5 , 1 , 7 ) .

  • A 9 𝑥 𝑦 + 7 𝑧 4 9 = 0
  • B 7 𝑥 3 5 𝑦 4 𝑧 2 8 = 0
  • C 4 𝑥 + 7 𝑧 6 9 = 0
  • D 7 𝑥 + 3 5 𝑦 4 𝑧 + 2 8 = 0
  • E 5 𝑥 𝑦 + 2 0 = 0

Q11:

Find the general equation of the plane that is perpendicular to the plane 8 𝑥 𝑦 + 6 𝑧 8 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 1 , 0 , 0 ) and ( 0 , 6 , 0 ) respectively.

  • A 8 𝑥 𝑦 + 6 𝑧 + 8 = 0
  • B 3 6 𝑥 + 6 𝑦 + 4 7 𝑧 + 3 6 = 0
  • C 8 𝑥 𝑦 + 6 𝑧 6 = 0
  • D 3 6 𝑥 + 6 𝑦 4 7 𝑧 + 3 6 = 0
  • E 3 6 𝑥 𝑦 + 6 𝑧 6 = 0

Q12:

Find the general equation of the plane that is perpendicular to the plane 6 𝑥 5 𝑦 7 𝑧 + 4 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 1 , 0 , 0 ) and ( 0 , 1 , 0 ) respectively.

  • A 6 𝑥 5 𝑦 7 𝑧 + 6 = 0
  • B 7 𝑥 7 𝑦 + 𝑧 7 = 0
  • C 6 𝑥 5 𝑦 7 𝑧 5 = 0
  • D 7 𝑥 + 7 𝑦 + 𝑧 + 7 = 0
  • E 7 𝑥 5 𝑦 7 𝑧 5 = 0

Q13:

Find the general equation of the plane that is perpendicular to the plane 3 𝑥 + 4 𝑦 2 𝑧 8 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 1 , 0 , 0 ) and ( 0 , 6 , 0 ) respectively.

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

Q14:

Find the general equation of the plane that is perpendicular to the plane 8 𝑥 5 𝑦 + 8 𝑧 + 5 = 0 and cuts the 𝑥 - and 𝑦 -axes at ( 4 , 0 , 0 ) and ( 0 , 3 , 0 ) respectively.

  • A 8 𝑥 5 𝑦 + 8 𝑧 + 3 2 = 0
  • B 6 𝑥 8 𝑦 𝑧 2 4 = 0
  • C 8 𝑥 5 𝑦 + 8 𝑧 1 5 = 0
  • D 6 𝑥 8 𝑦 + 𝑧 2 4 = 0
  • E 6 𝑥 5 𝑦 + 8 𝑧 1 5 = 0

Q15:

Find the general equation of the plane that passes through the point ( 1 , 7 , 6 ) and cuts off equal intercepts on the three coordinate axes.

  • A 𝑥 + 𝑦 + 𝑧 + 1 2 = 0
  • B 𝑥 + 7 𝑦 + 6 𝑧 = 0
  • C 𝑥 + 𝑦 + 𝑧 + 4 2 = 0
  • D 𝑥 + 𝑦 + 𝑧 1 2 = 0
  • E 𝑥 + 𝑦 + 𝑧 = 0

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