The area of the quadrilateral formed by the lines $4 x - 3 y - a = 0, 3 x - 4 y + a = 0$ , $4 x - 3 y - 3 a = 0$ and $3 x - 4 y + 2 a = 0$ is

\frac{2 a^{2}}{11} sq. units

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{2 a^{2}}{9} sq. units

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{a^{2}}{7} sq. units

No worries! We‘ve got your back. Try BYJU‘S free classes today!

\frac{2 a^{2}}{7} sq. units

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D $\frac{2 a^{2}}{7} sq. units$
Given lines
$L_{1} : 4 x - 3 y - a = 0 L_{2} : 3 x - 4 y + a = 0 L_{3} : 4 x - 3 y - 3 a = 0 L_{4} : 3 x - 4 y + 2 a = 0$
We know the distance between two parallel lines is
$p_{1} = ∣ ∣ ∣ ∣ \frac{- 3 a - (- a)}{\sqrt{4^{2} + 3^{2}}} ∣ ∣ ∣ ∣ = \frac{2 a}{5} units p_{2} = ∣ ∣ ∣ ∣ \frac{a - 2 a}{\sqrt{3^{2} + 4^{2}}} ∣ ∣ ∣ ∣ = \frac{a}{5} units$
Slope of the lines are
$m_{1} = \frac{3}{4}, m_{2} = \frac{4}{3}$
Now,
$tan θ = ∣ ∣ ∣ \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} ∣ ∣ ∣ \Rightarrow tan θ = ∣ ∣ ∣ ∣ ∣ ∣ \frac{\frac{3}{4} - \frac{4}{3}}{1 + 1} ∣ ∣ ∣ ∣ ∣ ∣ = \frac{7}{24} \Rightarrow sin θ = \frac{7}{25}$

Now, the area of the parallelogram
$= \frac{p_{1} p_{2}}{sin θ} = \frac{25 \times 2 a^{2}}{7 \times 25} = \frac{2 a^{2}}{7} sq. units$

Suggest Corrections

Similar questions

Q. Area of the quadrilateral formed by the lines

4 y - 3 x - a = 0, 3 y - 4 x + a = 0, 4 y - 3 x - 3 a = 0

3 y - 4 x + 2 a = 0

in sq.units is

Find the equation of circle circumscribing the quadrilateral formed by four lines $3 x + 4 y - 5 = 0$ , $4 x - 3 y - 5 = 0$ , $3 x + 4 y + 5 = 0$ and $4 x - 3 y + 5 = 0$

Q. Prove that the area of the parallelogram formed by the lines 3x − 4y + a = 0, 3x − 4y + 3a = 0, 4x − 3y − a = 0 and 4x − 3y − 2a = 0 is

\frac{2}{7} a^{2}

sq. units.

Q. The area of the parallelogram formed by the lines

3 x - 4 y + 1 = 0, 3 x - 4 y + 3 = 0, 4 x - 3 y - 1 = 0

and

4 x - 3 y - 2 = 0,

Q. For the quadrilateral formed by the lines

4 y - 3 x - 1 = 0, 3 y + 4 x + 1 = 0, 4 y - 3 x - 2 = 0

and

3 y + 4 x + 2 = 0

, which among the following statements are true

Join BYJU'S Learning Program

The area of the quadrilateral formed by the lines 4x−3y−a=0,3x−4y+a=0, 4x−3y−3a=0 and 3x−4y+2a=0 is

The area of the quadrilateral formed by the lines $4 x - 3 y - a = 0, 3 x - 4 y + a = 0$ , $4 x - 3 y - 3 a = 0$ and $3 x - 4 y + 2 a = 0$ is