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: Write the angle formula.
For the pair $ax^2 + 2hxy + by^2 = 0$, $\tan\theta = \left|\dfrac{2\sqrt{h^2 - ab}}{a+b}\right|$.

Step 2: Use the given angle.
Since $\theta = \dfrac{\pi}{4}$, $\tan\theta = 1$. Squaring gives $(a+b)^2 = 4(h^2 - ab)$.

Step 3: Solve for 4h squared.
\[ a^2 + 2ab + b^2 = 4h^2 - 4ab \Rightarrow 4h^2 = a^2 + 6ab + b^2 \]
\[ \boxed{a^2 + 6ab + b^2,\ \text{option 3}} \]