Question

If $(\text{m}+3\text{n})(3\text{m}+\text{n})=4\text{h}^2$, then the acute angle between the lines represented by $\text{m} \text{x}^2+2\text{h}\text{xy}+\text{n}\text{y}^2=0$ is

• A. $\frac{\pi^c}{3}$
• B. $\frac{\pi^c}{6}$
• C. $\tan^{-1}\left(\frac{3}{2}\right)$
• D. $\tan^{-1}\left(\frac{1}{2}\right)$

Hint:

Whenever the calculation reduces to a standard trigonometric value like $\tan \theta = \sqrt{3}$, you can instantly recall that the angle must be $60^\circ$ or $\frac{\pi}{3}$ radians.
Keep an eye out for perfect square structures like $(m+n)^2$ when working with homogeneous pair-of-lines problems to cancel variables out quickly.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given a specific algebraic constraint relating the coefficients $m, n,$ and $h$. We need to compute the acute angle $\theta$ between the pair of straight lines passing through the origin represented by the homogeneous quadratic equation $mx^2 + 2hxy + ny^2 = 0$.

Step 2: Key Formula or Approach:
The standard formula to find the acute angle $\theta$ between a pair of lines $ax^2 + 2hxy + by^2 = 0$ is given by:
$$\tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|$$ Adapting this formula directly to our given equation where $a = m$ and $b = n$, we have:
$$\tan \theta = \left| \frac{2\sqrt{h^2 - mn}}{m + n} \right|$$ We will use the given constraint equation to simplify the terms inside the square root.

Step 3: Detailed Explanation:
Let's first expand the given algebraic constraint:
$$(m + 3n)(3m + n) = 4h^2$$ $$3m^2 + mn + 9mn + 3n^2 = 4h^2$$ $$3m^2 + 10mn + 3n^2 = 4h^2$$ Now, let's isolate the term $4h^2 - 4mn$ to help match the numerator of our angle formula:
$$4h^2 - 4mn = (3m^2 + 10mn + 3n^2) - 4mn$$ $$4h^2 - 4mn = 3m^2 + 6mn + 3n^2$$ Factor out the common scalar value 3 from the right-hand expression:
$$4(h^2 - mn) = 3(m^2 + 2mn + n^2)$$ $$4(h^2 - mn) = 3(m + n)^2$$ Taking the square root on both sides yields:
$$2\sqrt{h^2 - mn} = \sqrt{3}|m + n|$$ Now, substitute this expression directly back into our core tangent angle formula:
$$\tan \theta = \left| \frac{2\sqrt{h^2 - mn}}{m + n} \right| = \frac{\sqrt{3}|m + n|}{|m + n|} = \sqrt{3}$$ Since $\tan \theta = \sqrt{3}$ for an acute angle, we find:
$$\theta = 60^\circ = \frac{\pi^c}{3}$$

Step 4: Final Answer:
The acute angle between the lines is $\frac{\pi^c}{3}$, which corresponds to option (A).