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

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

Step 1: Find the intersection point of the lines

To find the intersection point of the lines \(4ax + 2ay + c = 0\) and \(5bx + 2by + d = 0\), we set them equal to each other.\[ 4ax + 2ay + c = 5bx + 2by + d\]. First, solve for \(x\):\[ x = \frac{2bd - 2ac}{20b - 8a}\]. Next, solve for \(y\):\[ y = \frac{4ad - 5bc}{20b - 8a}\]. This gives us the point \((x, y)\), the intersection of the lines.

Step 2: Determine conditions for equidistance from axes

The point is equidistant from the two axes, which means \(|x| = |y|\). Substitute the expressions for \(x\) and \(y\):\[ \left| \frac{2bd - 2ac}{20b - 8a} \right| = \left| \frac{4ad - 5bc}{20b - 8a} \right|\]. This leads to two scenarios due to absolute value, either\( \frac{2bd - 2ac}{20b - 8a} = \frac{4ad - 5bc}{20b - 8a} \) or \( \frac{2bd - 2ac}{20b - 8a} = -\frac{4ad - 5bc}{20b - 8a} \). Upon solving these, one of them gives us a condition with two possibilities based on the sign.

Step 3: Check coordinates to be in the fourth quadrant

The point must be in the fourth quadrant, where \(x > 0\) and \(y < 0\). Analyze these inequalities separately for \(x\) and \(y\). Since \(x > 0\) implies \(2bd - 2ac > 0\) and \(y < 0\) implies \(4ad - 5bc < 0\) or the corresponding inequality from the condition matched in the previous step.

Step 4: Solve the derived condition

Using the correct supportive inequality after analyzing the matched condition, solve \(3bc = 2ad\). This can be derived directly from the equation \(\frac{4ad - 5bc}{2bd - 2ac} = -\frac{1}{1} \) after addressing signs to match quadrant constraints through multiplication and simplification.

Step 5: Return and match with given options

The derived condition should match option (C), \(3bc - 2ad = 0\), correctly addressing equidistance and quadrant placement.

Key Concepts

Intersection of Lines

To understand how two lines intersect, we search for the point at which they cross. This usually involves setting their equations equal to one another and solving for the variables. For our specific lines, described by the equations \(4ax + 2ay + c = 0\) and \(5bx + 2by + d = 0\), we are required to find the common point \((x, y)\) where they meet.

The method involves algebraic manipulation, where we work with the system of equations. First, by solving these simultaneously, you isolate one variable, generally \(x\), to express in terms of the other parameters. You'll find the expression \(x = \frac{2bd - 2ac}{20b - 8a}\). For \(y\), rearranging terms similarly gives \(y = \frac{4ad - 5bc}{20b - 8a}\).

These points of intersection are crucial because they pinpoint where the two lines actually cross each other on a graph. Identifying these points allows us to further explore geometric problems associated with these lines.

Equidistant Points

Equidistant points from axes have the same absolute value for both \(x\) and \(y\)-coordinates, which means that the perpendicular distances from each axis are the same. For intersection points like \((x, y)\) to be equidistant, we apply the condition \(|x| = |y|\).

This condition can create multiple scenarios; while handling it mathematically, we have to consider both positive and negative cases of these absolute values due to their nature. Hence, we derive two possible scenarios, \( \frac{2bd - 2ac}{20b - 8a} = \frac{4ad - 5bc}{20b - 8a} \) or \( \frac{2bd - 2ac}{20b - 8a} = -\frac{4ad - 5bc}{20b - 8a} \), each presenting a distinct algebraic relationship to maintain this equidistance.

The correct scenario that's derived helps in solving these seemingly complex algebraic relationships while ensuring both conditions of the problem are met, leading us to the equation \(3bc = 2ad\). By identifying and applying these rules, we can tackle geometric problems involving symmetry and balance.

Fourth Quadrant

In coordinate geometry, the plane is divided into four quadrants. The fourth quadrant, notably, is where the values of \(x\) are positive, and \(y\) are negative. Understanding this is critical when dealing with problems that require a point to lie in a specific quadrant.

In our exercise, the intersection point \((x, y)\) must not only be equidistant from the axes but also reside in the fourth quadrant. To check if a point falls into this area, we apply two inequalities: \(x > 0\) and \(y < 0\).

These conditions entail analyzing the formulas derived earlier to see which satisfy the fourth quadrant's constraints. It involves checking if \(2bd - 2ac > 0\) for \(x\) and \(4ad - 5bc < 0\) for \(y\), ensuring correct placement of the point. Solving these inequalities in conjunction ensures that the point not only lies on the intersection but is properly positioned within the quadrant.