Question

Assertion :If the angle between the two lines represented by $2x^2+5xy+3y^2+6x+7y+4=0$ is ${\tan }^{-1}\left(m\right)$, then value of $m=\frac{1}{5}$. Reason: The angle $\theta$ between the two lines $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ is calculated by ${\tan }^{-1}\frac{\left(2\sqrt{h^2-ab}\right)}{|a+b|}$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Solving the assertion gives us:

$2x^2+5xy+3y^2+6x+7y+4=0$

$\angle \theta ={\tan }^{-1}\frac{2\sqrt{h^2-ab}}{|a+b|}$

$\angle \theta ={\tan }^{-1}\frac{2\sqrt{\frac{25}{4}-6}}{|2+3|}$

$\angle \theta ={\tan }^{-1}\frac{1}{5}$

Now, in the reason we can see that the formula given is applicable in solving the assertion, therefore the reason is correct.

Hence both assertion and reason are correct and reason is correct explanation of assertion.