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Worksheet: General Equation of a Line through the Intersection Point of Two Lines

Q1:

Find the equation of the line perpendicular to βˆ’ 6 π‘₯ βˆ’ 𝑦 + 8 = 0 and passing through the intersection of the lines βˆ’ 4 π‘₯ βˆ’ 𝑦 βˆ’ 3 = 0 and βˆ’ 3 π‘₯ + 8 𝑦 βˆ’ 1 = 0 .

  • A 1 7 3 π‘₯ βˆ’ 1 5 8 𝑦 + 1 0 1 = 0
  • B 2 π‘₯ βˆ’ 3 𝑦 + 1 = 0
  • C 1 9 π‘₯ βˆ’ 3 9 𝑦 + 8 = 0
  • D 7 π‘₯ βˆ’ 4 2 𝑦 βˆ’ 1 = 0

Q2:

What is the equation of the line passing through 𝐴 ( βˆ’ 1 , 3 ) and the intersection of the lines 3 π‘₯ βˆ’ 𝑦 + 5 = 0 and 5 π‘₯ + 2 𝑦 + 3 = 0 ?

  • A 8 π‘₯ + 𝑦 + 8 = 0
  • B 2 3 π‘₯ + 7 𝑦 + 1 7 = 0
  • C 1 7 π‘₯ βˆ’ 2 𝑦 + 2 3 = 0

Q3:

What is the equation of the line passing through 𝐴 ( βˆ’ 3 , 5 ) and the intersection of the lines 4 π‘₯ + 2 𝑦 + 1 = 0 and 2 π‘₯ + 3 𝑦 βˆ’ 2 = 0 ?

  • A 6 π‘₯ + 5 𝑦 βˆ’ 1 = 0
  • B 1 8 π‘₯ + 2 3 𝑦 βˆ’ 1 3 = 0
  • C 3 0 π‘₯ + 1 7 𝑦 + 5 = 0

Q4:

What is the equation of the line passing through 𝐴 ( 2 , βˆ’ 1 ) and the intersection of the lines 2 π‘₯ + 𝑦 βˆ’ 4 = 0 and 2 π‘₯ βˆ’ 4 𝑦 βˆ’ 1 = 0 ?

  • A 4 π‘₯ βˆ’ 3 𝑦 βˆ’ 5 = 0
  • B 1 6 π‘₯ βˆ’ 2 7 𝑦 βˆ’ 1 1 = 0
  • C 1 6 π‘₯ + 3 𝑦 βˆ’ 2 9 = 0

Q5:

Find the equation of the straight line that is parallel to the 𝑦 -axis and passes through the point of intersection of the two straight lines 𝑦 = βˆ’ 3 and π‘₯ = 1 1 1 5 𝑦 .

  • A π‘₯ = βˆ’ 1 1 5 𝑦
  • B 𝑦 = βˆ’ 1 1 5
  • C π‘₯ = 1 1 5
  • D π‘₯ = βˆ’ 1 1 5

Q6:

Find the equation of the straight line that is parallel to the 𝑦 -axis and passes through the point of intersection of the two straight lines 𝑦 = 9 and π‘₯ = βˆ’ 5 2 𝑦 .

  • A π‘₯ = βˆ’ 4 5 2 𝑦
  • B 𝑦 = βˆ’ 4 5 2
  • C π‘₯ = 4 5 2
  • D π‘₯ = βˆ’ 4 5 2

Q7:

The function 𝑓 ( π‘₯ ) is represented by the line βƒ–     βƒ— 𝐴 𝐡 and the function 𝑔 ( π‘₯ ) is represented by the line βƒ–      βƒ— 𝑂 𝐴 where the coordinates of 𝐴 are ( 2 , 5 ) . Find the equations of 𝑓 ( π‘₯ ) and 𝑔 ( π‘₯ ) .

  • A 𝑓 ( π‘₯ ) = 5 , 𝑔 ( π‘₯ ) = 2 5 π‘₯
  • B 𝑓 ( π‘₯ ) = 2 , 𝑔 ( π‘₯ ) = 5 2 π‘₯
  • C 𝑓 ( π‘₯ ) = 2 , 𝑔 ( π‘₯ ) = 2 5 π‘₯
  • D 𝑓 ( π‘₯ ) = 5 , 𝑔 ( π‘₯ ) = 5 2 π‘₯
  • E 𝑓 ( π‘₯ ) = 5 , 𝑔 ( π‘₯ ) = 5 π‘₯ + 2

Q8:

Find the equation of the straight line that passes through the origin and the point of intersection of the two straight lines π‘₯ = βˆ’ 1 7 4 and 𝑦 = βˆ’ 5 .

  • A 𝑦 = 8 5 4 π‘₯
  • B 𝑦 = 1 7 2 0 π‘₯
  • C 𝑦 = βˆ’ 2 0 1 7 π‘₯
  • D 𝑦 = 2 0 1 7 π‘₯

Q9:

Find the equation of the straight line that passes through the origin and the point of intersection of the two straight lines π‘₯ = 1 4 and 𝑦 = βˆ’ 8 7 .

  • A 𝑦 = βˆ’ 2 7 π‘₯
  • B 𝑦 = βˆ’ 7 3 2 π‘₯
  • C 𝑦 = 3 2 7 π‘₯
  • D 𝑦 = βˆ’ 3 2 7 π‘₯

Q10:

Find the equation of the vector that is parallel to the -axis and passes through the point of intersection of the two straight lines and .

  • A
  • B
  • C
  • D

Q11:

Find the π‘₯ -coordinate of the point at which the straight line 3 π‘₯ + 9 𝑦 = 0 cuts the π‘₯ -axis.

  • A3
  • B9
  • C 1 9
  • D0

Q12:

Which of the following equations represents a line through the origin?

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

Q13:

Find the equation of the straight line that passes through the point of intersection of the two straight lines π‘₯ βˆ’ 8 𝑦 = 2 and βˆ’ 6 π‘₯ βˆ’ 8 𝑦 = 1 and parallel to the 𝑦 -axis.

  • A π‘₯ = βˆ’ 1 3 5 6
  • B π‘₯ = βˆ’ 1 7
  • C π‘₯ = 1 7 𝑦
  • D π‘₯ = 1 7

Q14:

Find the equation of the straight line that passes through the point of intersection of the two straight lines 2 π‘₯ βˆ’ 2 𝑦 = 1 and βˆ’ π‘₯ + 3 𝑦 = βˆ’ 8 and parallel to the 𝑦 -axis.

  • A π‘₯ = βˆ’ 1 5 4
  • B π‘₯ = 1 3 4
  • C π‘₯ = βˆ’ 1 3 4 𝑦
  • D π‘₯ = βˆ’ 1 3 4

Q15:

Find the equation of the straight line which passes through the point of intersection of the two lines βˆ’ 4 π‘₯ + 1 5 𝑦 = βˆ’ 1 5 and βˆ’ 4 π‘₯ + 3 𝑦 = 1 4 and is parallel to the straight line r = ⟨ 4 , 0 ⟩ + π‘˜ ⟨ 5 , βˆ’ 4 ⟩ .

  • A 𝑦 βˆ’ 5 4 π‘₯ = 2 0 3
  • B 𝑦 βˆ’ 5 4 π‘₯ = βˆ’ 2 0 3
  • C βˆ’ 𝑦 βˆ’ 4 5 π‘₯ + 2 0 3 = 0
  • D 𝑦 + 4 5 π‘₯ = βˆ’ 2 0 3

Q16:

Find the equation of the straight line which passes through the point of intersection of the two lines βˆ’ 1 3 π‘₯ βˆ’ 5 𝑦 = 1 4 and 2 π‘₯ + 1 5 𝑦 = βˆ’ 1 1 and is parallel to the straight line π‘₯ + 8 𝑦 = βˆ’ 1 4 .

  • A 𝑦 βˆ’ 1 8 π‘₯ = 2 1 5 2 9 6
  • B 𝑦 βˆ’ 1 8 π‘₯ = βˆ’ 2 1 5 2 9 6
  • C βˆ’ 𝑦 βˆ’ 8 π‘₯ + 2 1 5 2 9 6 = 0
  • D 𝑦 + 1 8 π‘₯ = βˆ’ 2 1 5 2 9 6

Q17:

Determine the equation of the line passing through the point of intersection of the two lines whose equations are 5 π‘₯ + 2 𝑦 = 0 and 3 π‘₯ + 7 𝑦 + 1 3 = 0 while making an angle of 1 3 5 ∘ with the positive 𝑦 -axis.

  • A 2 9 π‘₯ + 𝑦 βˆ’ 9 1 = 0
  • B 2 9 π‘₯ + 2 9 𝑦 + 3 9 = 0
  • C 2 9 π‘₯ + 2 9 𝑦 βˆ’ 3 9 = 0
  • D 2 9 π‘₯ βˆ’ 2 9 𝑦 βˆ’ 9 1 = 0

Q18:

Determine the equation of the line passing through the point of intersection of the two lines whose equations are 2 π‘₯ βˆ’ 𝑦 βˆ’ 7 = 0 and π‘₯ + 2 𝑦 βˆ’ 6 = 0 while making an angle of 4 5 ∘ with the positive 𝑦 -axis.

  • A π‘₯ βˆ’ 𝑦 βˆ’ 5 = 0
  • B π‘₯ βˆ’ 𝑦 βˆ’ 3 = 0
  • C π‘₯ βˆ’ 𝑦 + 3 = 0
  • D π‘₯ + 𝑦 βˆ’ 5 = 0

Q19:

Find the vector equation of the straight line that passes through the point of intersection of the two straight lines βˆ’ 8 π‘₯ βˆ’ 𝑦 = 7 and βˆ’ 5 π‘₯ βˆ’ 3 𝑦 = 2 and the point ( 1 2 , 8 ) .

  • A r = ( 1 2 , 8 ) + π‘˜ ( βˆ’ 1 3 , βˆ’ 7 )
  • B r = ( 1 2 , 8 ) + π‘˜ ( 7 , 1 3 )
  • C r = ( 1 3 , 7 ) + π‘˜ ( 1 2 , 8 )
  • D r = ⟨ 1 2 , 8 ⟩ + π‘˜ ⟨ 1 3 , 7 ⟩

Q20:

Find the vector equation of the straight line that passes through the point of intersection of the two straight lines 5 π‘₯ + 𝑦 = 6 and 2 π‘₯ + 1 3 𝑦 = 1 5 and the point ( 0 , βˆ’ 9 ) .

  • A r = ( 0 , βˆ’ 9 ) + π‘˜ ( βˆ’ 1 , βˆ’ 1 0 )
  • B r = ( 0 , βˆ’ 9 ) + π‘˜ ( 1 0 , 1 )
  • C r = ( 1 , 1 0 ) + π‘˜ ( 0 , βˆ’ 9 )
  • D r = ⟨ 0 , βˆ’ 9 ⟩ + π‘˜ ⟨ 1 , 1 0 ⟩