Key Concepts and Formulas

• Point of Intersection: The point $(x, y)$ lies on a line if and only if substituting $x$ and $y$ into the line's equation results in a true statement.
• Fourth Quadrant: A point $(x, y)$ lies in the fourth quadrant if $x > 0$ and $y < 0$.
• Equidistant from Axes: A point $(x, y)$ is equidistant from the x and y axes if $|x| = |y|$. In the fourth quadrant, this implies $x = -y$.

Step-by-Step Solution

Step 1: Define the intersection point based on the given conditions.

The problem states the intersection point lies in the fourth quadrant and is equidistant from both axes. Let this point be $(x, y)$. Since it's in the fourth quadrant, $x > 0$ and $y < 0$. Because it's equidistant from the axes, $|x| = |y|$. Therefore, $x = -y$. Let $x = k$, where $k > 0$. Then $y = -k$. Thus, the point of intersection is $(k, -k)$ where $k > 0$.

Step 2: Substitute the intersection point into the first equation.

The first equation is $4ax + 2ay + c = 0$. Substitute $x = k$ and $y = -k$ into this equation: $4a(k) + 2a(-k) + c = 0$ $4ak - 2ak + c = 0$ $2ak + c = 0$

Now, solve for $k$ in terms of $a$ and $c$: $2ak = -c$ $k = -\frac{c}{2a}$ Since $k > 0$, we know that $-\frac{c}{2a} > 0$, which implies $\frac{c}{a} < 0$. This means $a$ and $c$ have opposite signs.

Step 3: Substitute the intersection point into the second equation.

The second equation is $5bx + 2by + d = 0$. Substitute $x = k$ and $y = -k$ into this equation: $5b(k) + 2b(-k) + d = 0$ $5bk - 2bk + d = 0$ $3bk + d = 0$

Now, solve for $k$ in terms of $b$ and $d$: $3bk = -d$ $k = -\frac{d}{3b}$ Since $k > 0$, we know that $-\frac{d}{3b} > 0$, which implies $\frac{d}{b} < 0$. This means $b$ and $d$ have opposite signs.

Step 4: Eliminate $k$ to find a relationship between $a, b, c,$ and $d$.

We have two expressions for $k$: $k = -\frac{c}{2a}$ $k = -\frac{d}{3b}$ Since both expressions are equal to $k$, we can set them equal to each other: $-\frac{c}{2a} = -\frac{d}{3b}$

Now, we can cross-multiply to eliminate the fractions: $-3bc = -2ad$ $3bc = 2ad$

Finally, rearrange the terms to get the desired relationship: $3bc - 2ad = 0$

Common Mistakes & Tips

• Incorrect Quadrant: Confusing the signs in the fourth quadrant. Remember, $x > 0$ and $y < 0$.
• Sign Errors: Careless mistakes when substituting negative values. Double-check your work.
• Division by Zero: Always be mindful of the problem statement that $a, b, c,$ and $d$ are non-zero.

Summary

We used the given conditions to determine the coordinates of the intersection point as $(k, -k)$ where $k > 0$. We then substituted these coordinates into the equations of both lines, solved for $k$ in terms of the coefficients, and equated the two expressions for $k$ to derive the relationship $3bc - 2ad = 0$.

The final answer is $\boxed{3bc - 2ad = 0}$, which corresponds to option (A).