Question

If the acute angle between the lines given by $ax^2 + 2hxy + by^2 = 0$ is $\frac{\pi}{4}$, then $4h^2 =$

• A. $(a + 2b)(a + 3b)$
• B. $a^2 + 4ab + b^2$
• C. $a^2 + 6ab + b^2$
• D. $(a - 2b)(2 a + b)$

Hint:

Whenever an angle is $\frac{\pi}{4}$ ($45^\circ$), $\tan\theta = 1$, meaning the numerator and denominator inside the absolute brackets are equal before squaring: $|a+b| = 2\sqrt{h^2-ab}$. Squaring directly sets up $(a+b)^2 = 4h^2 - 4ab$, making it clean to expand and solve!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question presents a homogenous second-degree equation representing a pair of straight lines passing through the origin. Given that the acute angle $\theta$ between these lines is $\frac{\pi}{4}$, we need to find the algebraic expression equivalent to the term $4h^2$.

Step 2: Key Formula or Approach:
The standard formula to compute the acute angle $\theta$ between a pair of straight lines given by $ax^2 + 2hxy + by^2 = 0$ is: $$ \tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| $$

Step 3: Detailed Explanation: We are given that the acute angle $\theta = \frac{\pi}{4}$. Substituting this value into our angle formula: $$ \tan\left(\frac{\pi}{4}\right) = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| $$ Since $\tan\left(\frac{\pi}{4}\right) = 1$, the equation becomes: $$ 1 = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right| $$ To remove the absolute value bars and radical signs, let's square both sides of the equation: $$ 1^2 = \frac{4(h^2 - ab)}{(a + b)^2} $$ Cross-multiplying the denominator to the left side gives: $$ (a + b)^2 = 4(h^2 - ab) $$ Expanding the left side using the perfect square identity $(a+b)^2 = a^2 + 2ab + b^2$: $$ a^2 + 2ab + b^2 = 4h^2 - 4ab $$ To isolate the target term $4h^2$, let's shift the $-4ab$ term from the right side over to the left side: $$ 4h^2 = a^2 + 2ab + b^2 + 4ab $$ Combining the like terms ($2ab + 4ab = 6ab$) yields our final expression: $$ 4h^2 = a^2 + 6ab + b^2 $$

Step 4: Final Answer: The expression for $4h^2$ evaluates to $a^2 + 6ab + b^2$, which matches option (C).