Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A line is drawn from the point \(P(\alpha, \beta)\), making an angle \(\theta\) with the positive direction of \(x\)-axis, to meet the line \(a x+b y+c=0\) at \(Q\). The length of \(P Q\) is (A) \(-\frac{a \alpha+b \beta+c}{a \cos \theta+b \sin \theta}\) (B) \(\left|\frac{a \alpha+b \beta+c}{\sqrt{a^{2}+b^{2}}}\right|\) (C) \(\frac{a \alpha+b \beta+c}{a \cos \theta+b \sin \theta}\) (D) none of these

Step-by-Step Solution

Answer

Option (A) is correct.

Step 1: Understand the Problem

We need to find the length of a perpendicular from the point \(P(\alpha, \beta)\) to the line \(ax + by + c = 0\). This line from \(P\) to \(Q\) makes an angle \(\theta\) with the positive x-axis.

Step 2: Line Equation in Parametric Form

Express the line passing through point \(P(\alpha, \beta)\) at an angle \(\theta\) as a parametric equation: \(x = \alpha + t \cos \theta\), \(y = \beta + t \sin \theta\). Here, \(t\) is the parameter that measures the distance along the line from \(P\).

Step 3: Intersection with Given Line

Substitute the parametric equations of the line into the equation of the given line, \(ax + by + c = 0\):\[a(\alpha + t \cos \theta) + b(\beta + t \sin \theta) + c = 0.\]

Step 4: Solve for Parameter t

Solving the equation from Step 3:\[a\alpha + at \cos \theta + b\beta + bt \sin \theta + c = 0.\]This simplifies to:\[t(a \cos \theta + b \sin \theta) = -(a \alpha + b \beta + c).\]Thus, the parameter \(t\) is:\[t = \frac{-(a \alpha + b \beta + c)}{a \cos \theta + b \sin \theta}.\]

Step 5: Length of P to Q

The length \(PQ\) is given by the absolute value of \(t\) because \(t\) represents the distance along the line:\[PQ = |t| = \left|\frac{-(a \alpha + b \beta + c)}{a \cos \theta + b \sin \theta}\right|.\]

Step 6: Determine the Correct Option

Comparing the form we derived to the options given, the expression matches option (A):\(PQ = -\frac{a \alpha + b \beta + c}{a \cos \theta + b \sin \theta}.\) Here it represents a negative sign outside allowing direct correlation with (A), indicating option (A) is correct.

Key Concepts

Parametric Equations

Parametric equations are a way to express the coordinates of the points on a line based on a parameter, usually denoted as \( t \). They are particularly useful in describing the motion of a point along a path and are instrumental in coordinate geometry, especially for dealing with lines.Parametrically, any line can be described using initial points and directional vectors. In the given exercise, we start by describing the line that passes through the point \( P(\alpha, \beta) \) and makes an angle \( \theta \) with the x-axis. The parametric equations for this line are:

• \( x = \alpha + t \cos \theta \)
• \( y = \beta + t \sin \theta \)

Here, \( t \) represents the distance traveled along the line from the starting point \( P \). These equations show how position changes as \( t \) increases, also showing the effect of both angle \( \theta \) and the parameter \( t \) on the x and y coordinates.

Angle with x-axis

The angle a line makes with the x-axis is a crucial concept for understanding how that line is oriented in relation to the coordinate axes. It helps in determining the direction of the line, which is essential for converting between different forms of line equations.Given a line through point \( P(\alpha, \beta) \) making an angle \( \theta \) with the positive x-axis, this angle determines how changes along the line will affect the x and y coordinates. For this problem, the cosine of \( \theta \) affects the horizontal movement (change in \( x \)), and the sine of \( \theta \) affects the vertical movement (change in \( y \)).This concept becomes vital when substituting into the line equation \( ax + by + c = 0 \), as seen in the provided solution. It allows for the parametric representation to effectively translate orientation into movement along the line.

Perpendicular Distance Formula

The perpendicular distance formula is an essential tool in coordinate geometry. It calculates the shortest distance from a point to a given line. For a line described by \( ax + by + c = 0 \), and a point \( (x_1, y_1) \), the perpendicular distance \( D \) is:\[D = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}\]In this exercise, however, the problem instead looks at finding the distance along a particular direction, driven by angle \( \theta \). Once \( t \) is determined as:\[t = \frac{-(a\alpha + b\beta + c)}{a \cos \theta + b \sin \theta}\]We recognize that this expression directly gives the directed distance from \( P \) to \( Q \) along the line's trajectory. This solution reflects both directionality and perpendicular distance considerations, as it ensures that the line hits perfectly at right angles. Thus, confirming the correctness of option (A) in the choices provided.