Problem 41 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equations of the straight lines passing through \((-2,-7)\) and cutting an intercept of length three units between the straight lines \(4 x+3 y=12\) and \(4 x+3 y=\) 3 are (A) \(x+2=0, y+7=\frac{7}{24}(x+2)\) (B) \(x-2=0, y+7=-\frac{7}{24}(x+2)\) (C) \(x+2=0, y+7=-\frac{7}{24}(x+2)\) (D) \(x+2=0, y+7=-\frac{7}{12}(x+2)\)

Step-by-Step Solution

Answer

The correct line is not listed; line should be: \(x + 2 = 0\) or \(y + 7 = \frac{3}{4}(x + 2)\).

Step 1: Understand the Line Intercepts

We need to find a line that passes through the point \((-2, -7)\) and intersects the given lines \(4x + 3y = 12\) and \(4x + 3y = 3\) such that the intercept between them is 3 units long.

Step 2: Determine Slope of Given Lines

Both given lines \(4x + 3y = 12\) and \(4x + 3y = 3\) have the same slope because they are parallel. Transform to slope-intercept form: \(y = -\frac{4}{3}x + 4\) for the first line and \(y = -\frac{4}{3}x + 1\) for the second line.

Step 3: Identify the Intersection Segment

Compute the vertical distance between the parallel lines as the intercept length. The difference in the constant terms 4 and 1 is 3, and because lines are parallel same slope applies \(-\frac{4}{3}\). The vertical distance method (parallelism) confirms this intercept length is 3 units.

Step 4: Find Possible equations

The line that passes through point \((-2, -7)\) must be perpendicular to the lines \(4x + 3y = 12\) and \(4x + 3y = 3\). The perpendicular slope to \(-\frac{4}{3}\) is \(\frac{3}{4}\). The equation of the line through \((-2, -7)\) is \(y + 7 = \frac{3}{4}(x + 2)\). Solving gives \(4y = 3x - 5\).

Key Concepts

Parallel Lines

Parallel lines are an essential concept in analytic geometry. These are lines in a plane that never meet; that is, they have the same slope. In the context of our exercise, the equations \(4x + 3y = 12\) and \(4x + 3y = 3\) represent two parallel lines because both can be rearranged into the slope-intercept form as \(y = -\frac{4}{3}x + 4\) and \(y = -\frac{4}{3}x + 1\) respectively.

• They have the identical slope \(-\frac{4}{3}\), indicating they are parallel.
• Despite having different intercepts, they maintain the same gradient.

Parallel lines never intersect, meaning the lines extend in the same direction without crossing each other. The change in intercept between the two lines helps in determining a line's positioning relative to both lines. Understanding parallel lines is crucial in scenarios where consistent angles and spatial relationships across different equations are required.

Line Intercepts

Line intercepts refer to the points where a line cuts the axes. In the slope-intercept form of a linear equation \(y = mx + b\), the value \(b\) represents the y-intercept, while the x-intercept can be found by setting \(y=0\) and solving for \(x\). In the given problem of determining the intercept segment between the parallel lines \(4x + 3y = 12\) and \(4x + 3y = 3\), the intercept refers to the space the line crosses these parallel lines.

• The vertical distance or the space between the lines after rearranging in slope form helps judge the intercept length.
• Since both lines have no x-intercept changes, the intercept occurs along the y-axis.

Determining the intercept of length 3 units between the lines helps in plotting points or estimating where another line passing through a given point, like \((-2, -7)\), can interact with the fixed lines.

Perpendicular Slope

When it comes to perpendicular lines, their slopes are closely related. If two lines are perpendicular, the product of their slopes is \(-1\). This concept allows us to easily calculate one slope if the other is known. Given the slope \(-\frac{4}{3}\) of the parallel lines in the exercise, a perpendicular slope would be \(\frac{3}{4}\).

• This relationship is intrinsic, as perpendicular lines intersect at a right angle (90 degrees).
• The new line passing through \((-2, -7)\) with slope \(\frac{3}{4}\) ensures it cuts across the two parallel lines at this right angle.

To find the line equation using point-slope form with this perpendicular slope, substitute the point \((-2, -7)\) into the formula:\[y + 7 = \frac{3}{4}(x + 2)\]Solving this gives an understandable form to analyze how it perfectly intersects the two given parallel lines.