The area of a parallelogram formed by the lines ax - by - c- 0 is

The correct option is B 2c^2/ab The given lines are, ax ± by ± c = 0 ⇒x± c/a + y± c/b=1 The vertices are at A≡( ca, 0), C≡( ca, 0), B≡(0, ...

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Standard XI

Mathematics

Terminologies Related to an Ellipse

The area of a...

Question

The area of a parallelogram formed by the lines $a x \pm b y \pm c = 0$ is

\frac{c^{2}}{a b}

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\frac{2 c^{2}}{a b}

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\frac{c^{2}}{2 a b}

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None of these

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Solution

The correct option is B $\frac{2 c^{2}}{a b}$
The given lines are,
$a x \pm b y \pm c = 0$
$\Rightarrow \frac{x}{\pm c / a} + \frac{y}{\pm c / b} = 1$
The vertices are at $A \equiv (\frac{c}{a}, 0), C \equiv (- \frac{c}{a}, 0), B \equiv (0, \frac{c}{b}), D \equiv (0, - \frac{c}{b})$ . Therefore, the diagonals $A C$ and $B D$ of quadrilateral $A B C D$ are perpendicular( hint: since A,C are lies on x-axis and B,D are lies on y-axis).Hence, it is a rhombus whose area is given by,
$\frac{1}{2} \times A C \times B D = \frac{1}{2} \times \frac{2 c}{a} \times \frac{2 c}{b} = \frac{2 c^{2}}{a b}$

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Similar questions

Q. Show that the four lines

a x \pm b y \pm c = 0

enclose a rhombus whose area is

\frac{2 c^{2}}{a b}

Q. The area of a parallelogram formed by the lines

a x \pm b y \pm c = 0

, is?

Prove that the area of the parallelogram formed by the lines.

$a_{1} x + b_{1} y + c_{1} = 0, a_{1} x + b_{1} y + d_{1} = 0, a_{2} x + b_{2} y + c_{2} = 0, a_{2} x + b_{2} y + d_{2} = 0$ is $∣ ∣ \frac{(d_{1} - c_{1}) (d_{2} - c_{2})}{a_{1} b_{2} - a_{2} b_{1}} ∣ ∣$ sq. units.

Deduce the condition for these lines to form a rhombus.

Q. Prove that the area of the parallelogram formed by the lines a₁x + b₁y + c₁ = 0, a₁x + b₁y + d₁ = 0, a₂x + b₂y + c₂ = 0, a₂x + b₂y + d₂ = 0 is

|\frac{(d_{1} - c_{1}) (d_{2} - c_{2})}{a_{1} b_{2} - a_{2} b_{1}}|

sq. units.
Deduce the condition for these lines to form a rhombus.

Q. Assertion :The two straight lines given by the equation

(a^{2} - 3 b^{2}) x^{2} + 8 a b x y + (b^{2} - 3 a^{2}) y^{2} = 0

from with the line

a x + b y + c = 0

a triangle of area

\frac{c^{2}}{\sqrt{3} (a^{2} + b^{2})}

Reason: The triangles formed by the lines

(a^{2} - 3 b^{2}) x^{2} + 8 a b x y + (b^{2} - 3 a^{2}) y^{2} = 0

and

a x + b y + c = 0

is an equilateral triangle.

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The area of a parallelogram formed by the lines ax±by±c=0 is

The area of a parallelogram formed by the lines $a x \pm b y \pm c = 0$ is