Condition for a Straight Line Tangent to an Ellipse

The problem asks for the condition under which a given straight line is tangent to a specific ellipse. This involves finding the relationship between the parameters of the line and the ellipse for them to touch at exactly one point. Understanding the fundamental tangency condition for ellipses is crucial for solving this problem in coordinate geometry.

Understanding the Standard Tangency Condition

For an ellipse given by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), a straight line in the slope-intercept form, \( y = mx + c \), is tangent to the ellipse if and only if the following condition is satisfied:

\( c^2 = a^2 m^2 + b^2 \)

This tangency condition relates the y-intercept (\(c\)), the slope (\(m\)) of the line, and the semi-major and semi-minor axes (\(a\) and \(b\)) of the ellipse. It is a standard result used to determine if a line is tangent to an ellipse.

Converting the Given Straight Line Equation

The given straight line is \( x \cos \alpha + y \sin \alpha = p \). This equation is in the normal form. To apply the standard tangency condition \( c^2 = a^2 m^2 + b^2 \), we need to convert the given line equation into the slope-intercept form \( y = mx + c \).

Assuming \( \sin \alpha \neq 0 \), we can rearrange the equation to solve for \(y\):

\( y \sin \alpha = -x \cos \alpha + p \)

\( y = \left(-\frac{\cos \alpha}{\sin \alpha}\right) x + \frac{p}{\sin \alpha} \)

Identifying Slope and Intercept

By comparing this with the slope-intercept form \( y = mx + c \), we can identify the slope (\(m\)) and the y-intercept (\(c\)) of the given straight line:

• Slope, \( m = -\frac{\cos \alpha}{\sin \alpha} \)
• Y-intercept, \( c = \frac{p}{\sin \alpha} \)

Applying the Tangency Condition Formula

Now we substitute these values of \(m\) and \(c\) into the tangency condition \( c^2 = a^2 m^2 + b^2 \):

\( \left(\frac{p}{\sin \alpha}\right)^2 = a^2 \left(-\frac{\cos \alpha}{\sin \alpha}\right)^2 + b^2 \)

\( \frac{p^2}{\sin^2 \alpha} = a^2 \frac{\cos^2 \alpha}{\sin^2 \alpha} + b^2 \)

Simplifying to Find the Condition for Tangency

To find the condition on \(p\), we simplify the equation. Multiply both sides by \( \sin^2 \alpha \) (assuming \( \sin \alpha \neq 0 \)):

\( p^2 = a^2 \cos^2 \alpha + b^2 \sin^2 \alpha \)

This equation represents the required condition for the straight line \( x \cos \alpha + y \sin \alpha = p \) to be tangent to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). This condition for tangency is a fundamental result in the study of conic sections.

Considering Special Cases

The derivation assumed \( \sin \alpha \neq 0 \). Let's quickly check the cases where \( \sin \alpha = 0 \) or \( \cos \alpha = 0 \).

• If \( \sin \alpha = 0 \), then \( \cos \alpha = \pm 1 \). The line is \( \pm x = p \), or \( x = \pm p \). These are vertical tangent lines at \( x = \pm a \). The condition is \( p^2 = a^2 \). Our formula gives \( p^2 = a^2(\pm 1)^2 + b^2(0)^2 = a^2 \). The formula holds.
• If \( \cos \alpha = 0 \), then \( \sin \alpha = \pm 1 \). The line is \( \pm y = p \), or \( y = \pm p \). These are horizontal tangent lines at \( y = \pm b \). The condition is \( p^2 = b^2 \). Our formula gives \( p^2 = a^2(0)^2 + b^2(\pm 1)^2 = b^2 \). The formula holds.

The derived condition \( p^2 = a^2 \cos^2 \alpha + b^2 \sin^2 \alpha \) is therefore valid for all values of \( \alpha \).

Comparing with Options

We found the condition for the straight line to be tangent to the ellipse is \( p^2 = a^2 \cos^2 \alpha + b^2 \sin^2 \alpha \).

Let's compare this with the given options:

1. \(p^2 = \frac{a^2b^2}{4}\)
2. \(p^2 = a^2 \cos^2 \alpha - b^2 \sin^2 \alpha\)
3. \(p^2 = a^2 \cos^2 \alpha + b^2 \sin^2 \alpha\)
4. None of these

Option 3 matches our derived condition.

Conclusion

The condition for the straight line \( x \cos \alpha + y \sin \alpha = p \) to be tangent to the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is \( p^2 = a^2 \cos^2 \alpha + b^2 \sin^2 \alpha \). This condition for tangency is a key result in coordinate geometry for conic sections like the ellipse.