Q. 65.0( 2 Votes )
Prove that the family of lines represented by x(1 + λ) + y(2 – λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point.
Answer :
Given:
Lines represented by x(1 + λ) + y(2 – λ) + 5 = 0, λ being arbitrary
To prove:
The family of lines represented by x(1 + λ) + y(2 – λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point.
Explanation:
The given family of lines can be written as
x + 2y + 5 + λ (x − y) = 0
This line is of the form L1 + λL2 = 0, which passes through the intersection of L1 = 0 and L2 = 0.
⇒ x + 2y + 5 = 0
⇒ x − y = 0
Now, solving the lines:This is a fixed point.
Hence proved.
Rate this question :






















If a + b + c = 0, then the family of lines 3ax + by + 2c = 0 pass through fixed point
RD Sharma - MathematicsFind the equation of a straight line passing through the point of intersection of x + 2y + 3 = 0 and 3x + 4y + 7 = 0 and perpendicular to the straight line x – y + 9 = 0.
RD Sharma - MathematicsFind the equation of a straight line through the point of intersection of the lines 4x – 3y = 0 and 2x – 5y + 3 = 0 and parallel to 4x + 5 y + 6 = 0.
RD Sharma - MathematicsProve that the family of lines represented by x(1 + λ) + y(2 – λ) + 5 = 0, λ being arbitrary, pass through a fixed point. Also, find the fixed point.
RD Sharma - MathematicsFind the equation of the straight line which passes through the point of intersection of the lines 3x – y = 5 and x + 3y = 1 and makes equal and positive intercepts on the axes.
RD Sharma - MathematicsFind the equations of the lines through the point of intersection of the lines x – 3y + 1 = 0 and 2x + 5y – 9 = 0 and whose distance from the origin is.
Find the equations of the lines through the point of intersection of the lines x – y + 1 = 0 and 2x – 3y + 5 = 0 whose distance from the point (3, 2) is .
Find the equation of the straight line drawn through the point of intersection of the lines x + y = 4 and 2x – 3y = 1 and perpendicular to the line cutting off intercepts 5, 6 on the axes.
RD Sharma - Mathematics