Straight Line Equation Derivation

This solution details the process of finding the equation for a straight line. The line must pass through a specific point and be perpendicular to another given line. We'll utilize coordinate geometry principles and trigonometric identities.

Point Analysis

We are given the following information about the line we need to find:

• Point of Passage: The line goes through the point \((-\sin\theta, \cos\theta)\). Let's denote this point as \((x_1, y_1)\), so \(x_1 = -\sin\theta\) and \(y_1 = \cos\theta\).
• Reference Line: The line is perpendicular to \(x\cos\theta + y\sin\theta = 9\).

Given Line Slope Calculation

The standard form of a linear equation is \(Ax + By + C = 0\). The slope (\(m\)) of a line in this form is calculated as \(m = -\frac{A}{B}\).

For the reference line, \(x\cos\theta + y\sin\theta = 9\), we identify the coefficients:

• \(A = \cos\theta\)
• \(B = \sin\theta\)

Let the slope of this given line be \(m_1\). Using the slope formula:

\(m_1 = -\frac{\cos\theta}{\sin\theta}\)

Perpendicular Line Slope Determination

A key property of perpendicular lines is that the product of their slopes equals \(-1\). Let the slope of the required line be \(m_2\). The condition for perpendicularity is:

\(m_1 \times m_2 = -1\)

Substitute the calculated slope \(m_1\):

\(\left(-\frac{\cos\theta}{\sin\theta}\right) \times m_2 = -1\)

Now, solve for \(m_2\):

\(m_2 = \frac{-1}{-\frac{\cos\theta}{\sin\theta}} = \frac{\sin\theta}{\cos\theta}\)

Point-Slope Form Application

The point-slope form is a useful way to find the equation of a line when you know a point \((x_1, y_1)\) on the line and its slope (\(m\)). The formula is: \(y - y_1 = m(x - x_1)\).

We have the point \((x_1, y_1) = (-\sin\theta, \cos\theta)\) and the slope \(m_2 = \frac{\sin\theta}{\cos\theta}\). Plugging these into the formula:

\(y - \cos\theta = \left(\frac{\sin\theta}{\cos\theta}\right) (x - (-\sin\theta))\)

Simplify the expression inside the parenthesis:

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

Equation Simplification Process

To get rid of the fraction, we multiply both sides of the equation by \(\cos\theta\). We assume \(\cos\theta \neq 0\) for the slope to be defined.

\(\cos\theta \times (y - \cos\theta) = \sin\theta \times (x + \sin\theta)\)

Expand both sides:

\(y\cos\theta - \cos^2\theta = x\sin\theta + \sin^2\theta\)

Rearrange the terms to match the standard form \(Ax + By + C = 0\) or similar structures found in the options:

\(x\sin\theta - y\cos\theta + \sin^2\theta + \cos^2\theta = 0\)

Utilize the fundamental trigonometric identity \(\sin^2\theta + \cos^2\theta = 1\). Substituting this identity:

\(x\sin\theta - y\cos\theta + 1 = 0\)

Equation Result

The derived equation for the straight line is \(x\sin\theta - y\cos\theta + 1 = 0\). This equation satisfies both conditions: passing through the specified point and being perpendicular to the given line.