Question

Assertion :The two straight lines given by the equation $(a^2-3b^2)x^2+8abxy+(b^2-3a^2)y^2=0$ from with the line $ax+by+c=0$ a triangle of area $\frac{c^2}{\sqrt{3}(a^2+b^2)}$ Reason: The triangles formed by the lines $(a^2-3b^2)x^2+8abxy+(b^2-3a^2)y^2=0$ and $ax+by+c=0$ is an equilateral triangle.

• A. Assertion is True, reason is True ; reason is a correct explanation for assertion
• B. Assertion is True, reason is True ; reason is NOT a correct explanation for assertion
• C. Assertion is True, reason is False
• D. Assertion is False, reason is True

Answer 1

The correct option is A Assertion is True, reason is True ; reason is a correct explanation for assertion

Factorizing the given equation, we obtain

$(a-\sqrt{3}b)x+(b+\sqrt{3}a)y=0$ ...(1)

and $(a+\sqrt{3}b)x+(b-\sqrt{3}a)y=0$ ...(2)

We call these lines as $OA$ and $OB,$ respectively.

Now, $ax+by+c=0$ ...(3)

is the equation of the line $AB.$

The angle between the lines (1) and (3) is given by

$\tan {\theta }_1=\frac{\frac{a}{b}-\frac{a-\sqrt{3}b}{b+\sqrt{3}a}}{1+\left(\frac{a}{b}\right)\left(\frac{a-\sqrt{3}b}{b+\sqrt{3}a}\right)}$$=\frac{ab-\sqrt{3}{a}^2-ab+\sqrt{3}{b}^2}{b^2+\sqrt{3}ab+a^2-\sqrt{3}ab}$$=\frac{\sqrt{3}(a^2-b^2)}{(a^2+b^2)}=\sqrt{3}.$

$\therefore {\theta }_1={60}^0.$

The angle between the lines (2) and (3) is given by

$\tan {\theta }_2=\frac{\frac{a+\sqrt{3}b}{b-\sqrt{3}a}-\frac{a}{b}}{1+\frac{a}{b}\left(\frac{a+\sqrt{3}b}{b-\sqrt{3}a}\right)}=\frac{\sqrt{3}(a^2+b^2)}{(a^2+b^2)}=\sqrt{3}.$

$\therefore {\theta }_2={60}^0$

Since ${\theta }_1+{\theta }_2={120}^0,$ therefore, the angle between (1) and (2) must be ${60}^0.$

Hence, $△OAB$ is equilateral.

Now, area of $△OAB=\frac{1}{2}\times OL\times AB$$=OL\times AL=OL\times OL\cot {60}^0$ $(\because \tan {60}^0=\frac{OL}{AL})$

$=\frac{1}{\sqrt{3}}{OL}^2=\frac{1}{\sqrt{3}}{\left[\frac{c}{\sqrt{a^2+b^2}}\right]}^2$

$(\because OL$ is the $\perp$ distance)

from $O$ to the line (3)

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

Thus the given reason is the correct explanation of the statement.