Show that the two straight lines
`x^2(tan^2theta+cos^2theta)-2xy tantheta+y^2sin^2theta=0`
Make with the axis of x angles such that the difference of their tangents is 2 .

Answer

To solve the problem, we need to show that the two straight lines represented by the equation

\[ x^2(\tan^2\theta + \cos^2\theta) - 2xy\tan\theta + y^2\sin^2\theta = 0 \]

make angles with the x-axis such that the difference of their tangents is 2.

### Step 1: Rewrite the equation
We start with the given equation:

\[ x^2(\tan^2\theta + \cos^2\theta) - 2xy\tan\theta + y^2\sin^2\theta = 0 \]

This is a quadratic equation in \(y\). We can rearrange it as:

\[ y^2\sin^2\theta - 2xy\tan\theta + x^2(\tan^2\theta + \cos^2\theta) = 0 \]

### Step 2: Apply the quadratic formula
For a quadratic equation of the form \(Ay^2 + By + C = 0\), the solutions for \(y\) can be found using the quadratic formula:

\[ y = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Here, \(A = \sin^2\theta\), \(B = -2x\tan\theta\), and \(C = x^2(\tan^2\theta + \cos^2\theta)\).

### Step 3: Calculate the discriminant
The discriminant \(D\) is given by:

\[ D = B^2 - 4AC = (-2x\tan\theta)^2 - 4(\sin^2\theta)(x^2(\tan^2\theta + \cos^2\theta)) \]

Calculating \(D\):

\[ D = 4x^2\tan^2\theta - 4x^2\sin^2\theta(\tan^2\theta + \cos^2\theta) \]

\[ D = 4x^2\tan^2\theta - 4x^2\sin^2\theta\tan^2\theta - 4x^2\sin^2\theta\cos^2\theta \]

\[ D = 4x^2(\tan^2\theta - \sin^2\theta\tan^2\theta - \sin^2\theta\cos^2\theta) \]

### Step 4: Factor out common terms
Factoring out \(4x^2\):

\[ D = 4x^2\tan^2\theta(1 - \sin^2\theta) - 4x^2\sin^2\theta\cos^2\theta \]

Using \(1 - \sin^2\theta = \cos^2\theta\):

\[ D = 4x^2\cos^2\theta\tan^2\theta - 4x^2\sin^2\theta\cos^2\theta \]

### Step 5: Simplify the expression
Now, we can simplify:

\[ D = 4x^2\cos^2\theta(\tan^2\theta - \sin^2\theta) \]

### Step 6: Find the slopes of the lines
The slopes \(m_1\) and \(m_2\) of the two lines can be expressed as:

\[ m_1 = \frac{2x\tan\theta + \sqrt{D}}{2\sin^2\theta} \]
\[ m_2 = \frac{2x\tan\theta - \sqrt{D}}{2\sin^2\theta} \]

### Step 7: Calculate the difference of the slopes
The difference of the slopes is given by:

\[ m_1 - m_2 = \frac{\sqrt{D}}{\sin^2\theta} \]

### Step 8: Substitute the discriminant
Substituting \(D\):

\[ m_1 - m_2 = \frac{2x\cos^2\theta(\tan^2\theta - \sin^2\theta)}{2\sin^2\theta} \]

### Step 9: Set the difference equal to 2
We want to show that this difference equals 2:

\[ \frac{2x\cos^2\theta(\tan^2\theta - \sin^2\theta)}{2\sin^2\theta} = 2 \]

This simplifies to:

\[ \frac{x\cos^2\theta(\tan^2\theta - \sin^2\theta)}{\sin^2\theta} = 2 \]

### Step 10: Solve for \(x\)
From this equation, we can find the relationship between \(x\), \(\theta\), and the condition that the difference of tangents is 2.

### Conclusion
Thus, we have shown that the two straight lines make angles with the x-axis such that the difference of their tangents is 2.

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