Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11
Advertisement Remove all ads

Find the equation of the straight line passing through the point of intersection of 2x + y − 1 = 0 and x + 3y − 2 = 0 and making with the coordinate axes a triangle of area \[\frac{3}{8}\] sq. units. - Mathematics

Answer in Brief

Find the equation of the straight line passing through the point of intersection of 2x + y − 1 = 0 and x + 3y − 2 = 0 and making with the coordinate axes a triangle of area \[\frac{3}{8}\] sq. units.

Advertisement Remove all ads

Solution

The equation of the straight line passing through the point of intersection of 2x + y − 1 = 0 and x + 3y − 2 = 0 is given below:
2x + y − 1 + λ (x + 3y − 2) = 0

\[\Rightarrow\] (2 + λ)x + (1 + 3λ)y − 1 − 2λ = 0 

\[\Rightarrow \frac{x}{\frac{1 + 2\lambda}{2 + \lambda}} + \frac{y}{\frac{1 + 2\lambda}{1 + 3\lambda}} = 1\]

So, the points of intersection of this line with the coordinate axes are \[\left( \frac{1 + 2\lambda}{2 + \lambda}, 0 \right) \text { and } \left( 0, \frac{1 + 2\lambda}{1 + 3\lambda} \right)\].

It is given that the required line makes an area of \[\frac{3}{8}\] square units with the coordinate axes.

\[\frac{1}{2}\left| \frac{1 + 2\lambda}{2 + \lambda} \times \frac{1 + 2\lambda}{1 + 3\lambda} \right| = \frac{3}{8}\]

\[ \Rightarrow 3\left| 3 \lambda^2 + 7\lambda + 2 \right| = 4\left| 4 \lambda^2 + 4\lambda + 1 \right|\]

\[ \Rightarrow 9 \lambda^2 + 21\lambda + 6 = 16 \lambda^2 + 16\lambda + 4\]

\[ \Rightarrow 7 \lambda^2 - 5\lambda - 2 = 0\]

\[ \Rightarrow \lambda = 1, - \frac{2}{7}\]

Hence, the equations of the required lines are

\[3x + 4y - 1 - 2 = 0 \text { and } \left( 2 - \frac{2}{7} \right)x + \left( 1 - \frac{6}{7} \right)y - 1 + \frac{4}{7} = 0\]

\[ \Rightarrow 3x + 4y - 3 = 0 \text { and } 12x + y - 3 = 0\]

Concept: Straight Lines - Equation of Family of Lines Passing Through the Point of Intersection of Two Lines
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.19 | Q 8 | Page 131
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×