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Worksheet: Equation of a Plane in Space

Q1:

Given that the plane 2 π‘₯ + 6 𝑦 + 2 𝑧 = 1 8 intersects the coordinate axes π‘₯ , 𝑦 , and 𝑧 at the points 𝐴 , 𝐡 , and 𝐢 , respectively, find the area of β–³ 𝐴 𝐡 𝐢 .

  • 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

Q2:

Determine the general equation of the plane that intersects the negative π‘₯ -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 𝐢 ( 9 , βˆ’ 4 , βˆ’ 4 ) .

  • A 1 1 π‘₯ βˆ’ 4 𝑦 βˆ’ 4 𝑧 + 1 2 = 0
  • B 1 2 π‘₯ βˆ’ 4 1 𝑦 βˆ’ 8 𝑧 βˆ’ 2 4 = 0
  • C 2 π‘₯ + 3 𝑧 βˆ’ 6 = 0
  • D 1 2 π‘₯ + 4 1 𝑦 βˆ’ 8 𝑧 + 2 4 = 0
  • E 9 π‘₯ βˆ’ 4 𝑦 βˆ’ 7 𝑧 + 1 8 = 0

Q3:

Determine the general equation of the plane that intersects the negative π‘₯ -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 𝐢 ( βˆ’ 6 , 1 , βˆ’ 2 ) .

  • A βˆ’ π‘₯ + 𝑦 βˆ’ 2 𝑧 βˆ’ 1 2 = 0
  • B 6 π‘₯ βˆ’ 1 6 𝑦 + 5 𝑧 βˆ’ 3 0 = 0
  • C 5 π‘₯ βˆ’ 6 𝑧 + 1 8 = 0
  • D 6 π‘₯ + 1 6 𝑦 + 5 𝑧 + 3 0 = 0
  • E βˆ’ 6 π‘₯ + 𝑦 + 4 𝑧 βˆ’ 3 0 = 0

Q4:

Determine the general equation of the plane that intersects the positive π‘₯ -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 𝐢 ( 4 , 4 , βˆ’ 3 ) .

  • A βˆ’ 2 π‘₯ + 4 𝑦 βˆ’ 3 𝑧 βˆ’ 6 = 0
  • B 4 π‘₯ + 7 𝑦 βˆ’ 1 2 𝑧 + 2 4 = 0
  • C βˆ’ 3 π‘₯ βˆ’ 𝑧 + 9 = 0
  • D 4 π‘₯ βˆ’ 7 𝑦 βˆ’ 1 2 𝑧 βˆ’ 2 4 = 0
  • E 4 π‘₯ + 4 𝑦 βˆ’ 𝑧 βˆ’ 2 4 = 0

Q5:

Determine the general equation of the plane that intersects the negative π‘₯ -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 𝐢 ( 6 , βˆ’ 6 , βˆ’ 9 ) .

  • A 4 π‘₯ βˆ’ 2 𝑦 βˆ’ 3 𝑧 βˆ’ 6 = 0
  • B βˆ’ 2 π‘₯ βˆ’ 5 𝑦 βˆ’ 6 𝑧 + 1 2 = 0
  • C 3 π‘₯ βˆ’ 𝑧 βˆ’ 2 7 = 0
  • D βˆ’ 2 π‘₯ + 5 𝑦 βˆ’ 6 𝑧 βˆ’ 1 2 = 0
  • E 6 π‘₯ βˆ’ 6 𝑦 βˆ’ 7 𝑧 + 3 6 = 0

Q6:

Determine the general equation of the plane that intersects the negative π‘₯ -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 𝐢 ( 5 , βˆ’ 1 , 7 ) .

  • A 9 π‘₯ βˆ’ 𝑦 + 7 𝑧 βˆ’ 4 9 = 0
  • B 7 π‘₯ βˆ’ 3 5 𝑦 βˆ’ 4 𝑧 βˆ’ 2 8 = 0
  • C 4 π‘₯ + 7 𝑧 βˆ’ 6 9 = 0
  • D 7 π‘₯ + 3 5 𝑦 βˆ’ 4 𝑧 + 2 8 = 0
  • E 5 π‘₯ βˆ’ 𝑦 + 2 0 = 0

Q7:

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

  • A βˆ’ 6 π‘₯ + 3 𝑦 + 4 𝑧 + 3 0 = 0
  • B 4 π‘₯ βˆ’ 2 0 𝑦 βˆ’ 9 𝑧 + 2 0 = 0
  • C βˆ’ 6 π‘₯ + 3 𝑦 + 4 𝑧 βˆ’ 3 = 0
  • D 4 π‘₯ + 2 0 𝑦 βˆ’ 9 𝑧 βˆ’ 2 0 = 0
  • E 4 π‘₯ + 3 𝑦 + 4 𝑧 βˆ’ 3 = 0

Q8:

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

  • A βˆ’ 8 π‘₯ βˆ’ 𝑦 + 6 𝑧 + 8 = 0
  • B 3 6 π‘₯ + 6 𝑦 + 4 7 𝑧 + 3 6 = 0
  • C βˆ’ 8 π‘₯ βˆ’ 𝑦 + 6 𝑧 βˆ’ 6 = 0
  • D βˆ’ 3 6 π‘₯ + 6 𝑦 βˆ’ 4 7 𝑧 + 3 6 = 0
  • E 3 6 π‘₯ βˆ’ 𝑦 + 6 𝑧 βˆ’ 6 = 0

Q9:

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

  • A 6 π‘₯ βˆ’ 5 𝑦 βˆ’ 7 𝑧 + 6 = 0
  • B 7 π‘₯ βˆ’ 7 𝑦 + 𝑧 βˆ’ 7 = 0
  • C 6 π‘₯ βˆ’ 5 𝑦 βˆ’ 7 𝑧 βˆ’ 5 = 0
  • D 7 π‘₯ + 7 𝑦 + 𝑧 + 7 = 0
  • E 7 π‘₯ βˆ’ 5 𝑦 βˆ’ 7 𝑧 βˆ’ 5 = 0

Q10:

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

  • A 3 π‘₯ + 4 𝑦 βˆ’ 2 𝑧 + 3 = 0
  • B 6 π‘₯ βˆ’ 𝑦 + 1 1 𝑧 βˆ’ 6 = 0
  • C 3 π‘₯ + 4 𝑦 βˆ’ 2 𝑧 + 2 4 = 0
  • D 6 π‘₯ + 𝑦 + 1 1 𝑧 + 6 = 0
  • E 6 π‘₯ + 4 𝑦 βˆ’ 2 𝑧 + 2 4 = 0

Q11:

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

  • A 8 π‘₯ βˆ’ 5 𝑦 + 8 𝑧 + 3 2 = 0
  • B 6 π‘₯ βˆ’ 8 𝑦 βˆ’ 𝑧 βˆ’ 2 4 = 0
  • C 8 π‘₯ βˆ’ 5 𝑦 + 8 𝑧 βˆ’ 1 5 = 0
  • D βˆ’ 6 π‘₯ βˆ’ 8 𝑦 + 𝑧 βˆ’ 2 4 = 0
  • E 6 π‘₯ βˆ’ 5 𝑦 + 8 𝑧 βˆ’ 1 5 = 0

Q12:

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

  • A π‘₯ + 𝑦 + 𝑧 βˆ’ 1 0 = 0
  • B 8 π‘₯ βˆ’ 9 𝑦 βˆ’ 9 𝑧 = 0
  • C π‘₯ + 𝑦 + 𝑧 βˆ’ 6 4 8 = 0
  • D π‘₯ + 𝑦 + 𝑧 + 1 0 = 0
  • E 8 π‘₯ + 𝑦 + 𝑧 = 0

Q13:

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

  • A π‘₯ + 𝑦 + 𝑧 + 1 2 = 0
  • B βˆ’ π‘₯ + 7 𝑦 + 6 𝑧 = 0
  • C π‘₯ + 𝑦 + 𝑧 + 4 2 = 0
  • D π‘₯ + 𝑦 + 𝑧 βˆ’ 1 2 = 0
  • E βˆ’ π‘₯ + 𝑦 + 𝑧 = 0

Q14:

Find the general equation of the plane that passes through the point ( βˆ’ 5 , βˆ’ 5 , 4 ) and cuts off equal intercepts on the three coordinate axes.

  • A π‘₯ + 𝑦 + 𝑧 βˆ’ 6 = 0
  • B βˆ’ 5 π‘₯ βˆ’ 5 𝑦 + 4 𝑧 = 0
  • C π‘₯ + 𝑦 + 𝑧 βˆ’ 1 0 0 = 0
  • D π‘₯ + 𝑦 + 𝑧 + 6 = 0
  • E βˆ’ 5 π‘₯ + 𝑦 + 𝑧 = 0

Q15:

Find the equation of the plane cutting the coordinate axes at 𝐴 , 𝐡 , and 𝐢 , given that the intersection point of the medians of β–³ 𝐴 𝐡 𝐢 is ( 𝑙 , π‘š , 𝑛 ) .

  • A 𝑙 π‘₯ + π‘š 𝑦 + 𝑛 𝑧 = 3
  • B 𝑙 π‘₯ + π‘š 𝑦 + 𝑛 𝑧 = 1
  • C π‘₯ 𝑙 + 𝑦 π‘š + 𝑧 𝑛 = 1
  • D π‘₯ 𝑙 + 𝑦 π‘š + 𝑧 𝑛 = 3
  • E π‘₯ + 𝑦 + 𝑧 = 𝑙 + π‘š + 𝑛