Problem 33 - A Complete Resource Book in Mathematics for JEE Main

Question

A line through \(A(-5,-4)\) meets the lines \(x+3 y+2=0\), \(2 x+y+4=0\) and \(x-y-5=0\) at the point \(B, C\) and \(D\), respectively. If \(\left(\frac{15}{A B}\right)^{2}+\left(\frac{10}{A C}\right)^{2}=\left(\frac{6}{A D}\right)^{2}\), the equa- tion of the line is (A) \(2 x+3 y+22=0\) (B) \(2 x-3 y+22=0\) (C) \(3 x+2 y+22=0\) (D) \(3 x-2 y+22=0\)

Step-by-Step Solution

Answer

(B) The equation of the line is \(2x - 3y + 22 = 0\).

Step 1: Identify line equation through point A

To find a line equation through point A(-5, -4), we'll use the general form of the line equation: \(y = mx + c\). Plug in the coordinates into the equation to find \(c\): \(-4 = m(-5) + c\).So, \(c = m(5) - 4\). The line equation becomes \(y = mx + 5m - 4\).

Step 2: Find points B, C, and D intersection coordinates

The given equations of lines are:1. \(x + 3y + 2 = 0\)2. \(2x + y + 4 = 0\)3. \(x - y - 5 = 0\)Set each equation equal to the form of line from Step 1 and solve for \(x\) and \(y\) intersection points B, C, and D. For example, substitute \(y\) from the line equation into each of these and solve for the \(x\) coordinate.

Step 3: Calculate distances AB, AC, and AD

Using the distance formula \[ AB = \sqrt{(x_B + 5)^2 + (y_B + 4)^2} \]\[ AC = \sqrt{(x_C + 5)^2 + (y_C + 4)^2} \]\[ AD = \sqrt{(x_D + 5)^2 + (y_D + 4)^2} \]calculate the distances from point A to the points B, C, and D once coordinates of B, C, and D are known.

Step 4: Apply given equation constraint

Take the constraint from the problem: \[\left(\frac{15}{AB}\right)^2 + \left(\frac{10}{AC}\right)^2 = \left(\frac{6}{AD}\right)^2\]Substitute the calculated distances into this equation and solve for \(m\) to determine the slope of the line through point A.

Step 5: Select correct line equation

Using the found slope \(m\), check which among the given options (A, B, C, D) satisfies the equation \(y = mx + 5m - 4\) and matches with one of the standard line equations given as a result choice.

Key Concepts

Line Intersection

Understanding line intersection is like finding the crossing point between two roads. When two lines intersect, they meet at a single point in the plane. To find this intersection point, you need to solve the equations of both lines simultaneously.
The given exercise includes lines with equations like:

• \(x + 3y + 2 = 0\)
• \(2x + y + 4 = 0\)
• \(x - y - 5 = 0\)

To find where each of these lines intersect another line passing through a point, say point \(A(-5, -4)\), you replace \(y\) in each given equation with \(mx + 5m - 4\). Next, solve for \(x\) and then substitute back to get \(y\). This gives you the exact coordinates where these lines intersect the given line.

Distance Formula

The distance formula helps you find the length between two points in a plane. Think of it like a ruler for the XY-plane. It's derived from the Pythagorean theorem and is represented as: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]In this exercise, you're tasked with finding distances like \(AB\), \(AC\), and \(AD\) between a fixed point \(A(-5, -4)\) and the intersection points \(B\), \(C\), \(D\) obtained earlier.

• Calculate \(AB\) using coordinates of \(B\) and \(A\)
• Calculate \(AC\) using coordinates of \(C\) and \(A\)
• Calculate \(AD\) using coordinates of \(D\) and \(A\)

Plug the coordinates into the distance formula, and you'll have the lengths needed to check the given equation constraint in the problem.

Slope of a Line

The slope of a line represents its steepness, or how much it goes up or down as you move along it. Visualize it as ski slope gradient - steep, flat, or downhill. Its mathematical representation is 'm' when you express a line in the form \(y = mx + c\). For instance, if a line passes through point \(A(-5, -4)\), then its equation becomes \(y = mx + 5m - 4\), needing \( m \) to find how steep the line is compared to the horizontal. You identify the correct slope by solving the problem's condition \[ \left(\frac{15}{AB}\right)^2 + \left(\frac{10}{AC}\right)^2 = \left(\frac{6}{AD}\right)^2 \]in Step 4, where \( AB, AC, \) and \( AD \) are computed as per the solutions. Solving for \( m \) determines the correct angle of the line through \( A \).

Equation of a Line

The equation of a line is a mathematical way to describe all the points that form that line on a plane. In general form, it's written as \(ax + by + c = 0\), where the constants \(a\), \(b\), and \(c\) define the line's orientation and location. In this exercise, given a point \(A(-5, -4)\) and a slope \(m\), the equation is rewritten in a form: \(y = mx + 5m - 4\). You derive the final line equation by substituting this slope, ensuring it matches one from the choices given (such as \(2x + 3y + 22 = 0\)).To find the correct option, verify which choice satisfies the computed line equation you have using your calculated \(m\). This will help in selecting the line that passes through point \(A\) and intersects each of the given lines at corresponding points \(B\), \(C\), and \(D\).