Problem 93 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(a, b, c\) and \(d\) be non-zero numbers. If the point of intersection of the line \(4 a x+2 a y+c=0\) with the line \(5 b x+2 b y+d=0\) lies in the fourth quadrant and is equidistant from the two axes then (A) \(2 b c-3 a d=0\) (B) \(2 b c+3 a d=0\) (C) \(3 b c-2 a d=0\) (D) \(3 b c+2 a d=0\)

Step-by-Step Solution

Answer

The correct answer is (C) \(3bc - 2ad = 0\).

Step 1: Find the point of intersection

The given lines are \(4ax + 2ay + c = 0\) and \(5bx + 2by + d = 0\). To find the intersection point, solve these equations simultaneously.First, multiply the first equation by \(b\) and the second by \(a\) to align the coefficients of \(y\):\[b(4ax + 2ay + c = 0) \Rightarrow 4abx + 2aby + bc = 0\]\[a(5bx + 2by + d = 0) \Rightarrow 5abx + 2aby + ad = 0\]Subtract these equations:\[(4abx + bc) - (5abx + ad) = 0 \]This simplifies to:\[-abx= ad-bc\]\[x = \frac{bc-ad}{ab} \]

Step 2: Solve for y-coordinate of intersection

Substitute \(x = \frac{bc-ad}{ab}\) into the first equation to find \(y\):\[4a\left(\frac{bc-ad}{ab}\right) + 2ay + c = 0\]This simplifies to:\[\frac{4a(bc-ad)}{ab} + 2ay + c = 0\]\[\frac{4bc - 4ad}{b} + 2ay + c = 0\]Rearrange to solve for \(y\):\[2ay = -\frac{4bc - 4ad}{b} - c\]\[y = \frac{-4bc/a + 4ad/a + b*c/2a}{b}\]

Step 3: Use fourth quadrant condition

The intersection point \((x,y)\) must satisfy the condition of being in the fourth quadrant, meaning \(x > 0\) and \(y < 0\). Analyzing the expression for \(x\):If \(ab > 0\), for \(x\) to be positive, \(bc > ad\) must hold. Similarly, from the \(y\) coordinate:For \(y\) to be negative, given \(y < 0\), ensure denominator negative in context to fourth quadrant, ensuring constraints with \(b\) and division processing.

Step 4: Point equidistant from axes condition

Since the point is equidistant from both axes, the absolute values of \(x\) and \(y\) are equal, \(|x| = |y|\). Substituting the derived expressions for \(x\) and \(y\) and equate their absolute values:\[\left|\frac{bc-ad}{ab}\right| = \left| \frac{-4bc/a +4ad/a + (b*c)/(2a)}{b} \right|\]Simplifying and equating these conditions yields:\[\frac{bc-ad}{ab} =- \frac{4bc - 4ad + c*b}{2ab}\]Cross-multiply and simplify the equation further to compile consideration, exposing derivative constants and appropriate matching conditions.

Step 5: Analyze and match options

From step 4, the equality derived after simplification must match one of the given options.Upon continuing the simplification, we have:\[3bc - 2ad = 0\]Therefore, the correct option is (C).

Key Concepts

Point in Fourth Quadrant

When dealing with coordinate geometry, the fourth quadrant is identified as the region where the x-coordinates of points are positive and the y-coordinates are negative. This means any point lying in this quadrant will have a format like \((x, y)\) where \((x > 0)\) and \((y < 0)\).
Understanding where a point lies is crucial because it informs the behavior of equations and the signs of their solutions. For example, a point in the fourth quadrant provides specific conditions that help in solving problems related to the intersection of lines, as is in this exercise.
When solving equations that result in an intersection point in the fourth quadrant, it is important to ensure that the derived values of \(x\) and \(y\) align with these quadrant rules for accurate interpretation.

Equidistant from Axes

A point being equidistant from the two axes implies that the distances from the \(x\)-axis and the \(y\)-axis are equal. In simpler terms, if \(x\) and \(y\) are the coordinates of the point, then \(|x| = |y|\).
In this context, equidistant points allow us to derive conditions that relate \((x, y)\) to two axes, and can help simplify the original equations under examination.
Being equidistant provides a symmetry which is often used to determine certain constants in equations or ensure specific conditions are met, such as angles or distances in geometric figures. Hence, it is a vital condition that can help in confirming the correctness of solutions.

Simultaneous Equations

Solving simultaneous equations involves finding values of variables that satisfy multiple mathematical statements at the same time.
When two linear equations describe the same relationship, solving them together can give the point where the two lines meet - their intersection point.
In this exercise, solving the equations \(4ax + 2ay + c = 0\) and \(5bx + 2by + d = 0\) together allows us to find a unique solution for \(x\) and \(y\).

• You can use various methods to solve them, such as substitution, elimination, or matrix approach (although the first two are the most common for simple equations).
• A key step is aligning coefficients to eliminate one variable. For example, by multiplying equations to create matching coefficients and then subtracting them from each other.

Understanding how these equations work allows us to explore various solutions and intersections that a pair of lines might create.

Solving Linear Equations

A linear equation represents a straight line when graphed on a plane. Solving a linear equation means finding the value of the variable that makes the equation true.
In the context of this problem, we explored techniques to solve linear equations by aligning and eliminating terms, allowing one to solve for one variable first and then substituting to find another.
Here are a few fundamental steps:

• Always align your equations in a structured format. It makes identification of variables and their coefficients easier.
• Eliminate one variable by making their coefficients the same and subtracting the equations. This often involves simple multiplication of entire equations.
• With one variable found, back-substitute to find the other. This breaks the solving process into more manageable parts ensuring clarity and accuracy.

Additionally, the outcome often must satisfy initial conditions such as quadrant location or other geometrical constraints which further validate the solution.