Question

If the two lines given by $ax^2 + 2hxy + by^2 = 0$ make inclinations $\alpha$ and $\beta$, then $\tan(\alpha + \beta) =$

• A. $\frac{h}{a+b}$
• B. $\frac{2h}{a+b}$
• C. $\frac{h}{a-b}$
• D. $\frac{2h}{a-b}$

Hint:

Don't confuse $\tan(\alpha + \beta)$ with the formula for the acute angle $\theta$ between the two lines! The angle between the lines is $\tan \theta = \left| \frac{2\sqrt{h^2 - ab}}{a + b} \right|$, which relies on the difference of the slopes ($m_1 - m_2$) rather than their sum.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The equation $ax^2 + 2hxy + by^2 = 0$ represents a pair of straight lines passing through the origin. We are given the angles of inclination ($\alpha$ and $\beta$) of these two lines, meaning their individual slopes are $\tan \alpha$ and $\tan \beta$. We need to find the value of $\tan(\alpha + \beta)$.

Step 2: Key Formula or Approach:
For the general homogeneous equation of second degree $ax^2 + 2hxy + by^2 = 0$, let the slopes of the two lines be $m_1$ and $m_2$.
The sum of the slopes is given by: $m_1 + m_2 = -\frac{2h}{b}$
The product of the slopes is given by: $m_1 m_2 = \frac{a}{b}$
The trigonometric identity for the tangent of a sum is:
$$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$

Step 3: Detailed Explanation:
Substitute $m_1 = \tan \alpha$ and $m_2 = \tan \beta$ into our sum and product formulas:
$$\tan \alpha + \tan \beta = -\frac{2h}{b}$$ $$\tan \alpha \tan \beta = \frac{a}{b}$$ Now, substitute these expressions into the trigonometric identity:
$$\tan(\alpha + \beta) = \frac{-\frac{2h}{b}}{1 - \frac{a}{b}}$$ To simplify the complex fraction, multiply the numerator and the denominator by $b$:
$$\tan(\alpha + \beta) = \frac{-2h}{b - a}$$ Multiply the numerator and denominator by $-1$ to rearrange the terms in the denominator into standard alphabetical order:
$$\tan(\alpha + \beta) = \frac{2h}{a - b}$$

Step 4: Final Answer:
The value of $\tan(\alpha + \beta)$ is $\frac{2h}{a-b}$, which corresponds to option (D).