Key Concepts and Formulas

• Intercept Form of a Line: A straight line that intercepts the $x$-axis at $(a,0)$ and the $y$-axis at $(0,b)$ can be conceptually represented as $\frac{x}{a} + \frac{y}{b} = 1$. The intercepts are $a$ and $b$ respectively.
• Area of a Triangle Formed by a Line with Coordinate Axes: If a line has x-intercept 'a' and y-intercept 'b', the area of the triangle formed by this line and the coordinate axes is given by $\frac{1}{2} |a \cdot b|$.
• Normal Form (Perpendicular Form) of a Line: The equation of a straight line whose perpendicular distance from the origin is $p$ and the perpendicular makes an angle $\alpha$ with the positive $x$-axis is $x \cos \alpha + y \sin \alpha = p$. Importantly, $p$ is always a positive distance.

Step-by-Step Solution

Step 1: Determine the intercepts of the line L in terms of b and c.

The given equation of the line L is $x + by + c = 0$. To find the $x$ and $y$ intercepts, we set $y=0$ and $x=0$ respectively.

• x-intercept: Setting $y=0$ in the equation $x + by + c = 0$, we get $x + b(0) + c = 0$, which simplifies to $x = -c$. Therefore, the $x$-intercept is $a = -c$.
• y-intercept: Setting $x=0$ in the equation $x + by + c = 0$, we get $0 + by + c = 0$, which simplifies to $y = -\frac{c}{b}$. Thus, the $y$-intercept is $b' = -\frac{c}{b}$. (We use $b'$ to distinguish from the coefficient $b$ in the line's equation).

Step 2: Use the area information to establish a relationship between b and c.

The area of the triangle formed by the line L and the coordinate axes is given as 48 square units. We use the formula for the area of a triangle formed by a line with the coordinate axes:

$Area = \frac{1}{2} |a \cdot b'| = 48$

Substituting the values of the intercepts $a = -c$ and $b' = -\frac{c}{b}$, we get:

$\frac{1}{2} \left| (-c) \left(-\frac{c}{b}\right) \right| = 48$

$\frac{1}{2} \left| \frac{c^2}{b} \right| = 48$

Multiplying both sides by 2 gives:

$\left| \frac{c^2}{b} \right| = 96$

Since $c^2$ is always non-negative, we can rewrite this as:

$\frac{c^2}{|b|} = 96 \quad (*)$

This equation links $b$ and $c$.

Step 3: Use the perpendicular information to determine the value of b and the relationship between c and p.

The perpendicular drawn from the origin to the line L makes an angle of $45^{\circ}$ with the positive $x$-axis. This corresponds to the angle $\alpha$ in the normal form, hence $\alpha = 45^{\circ}$. The normal form of the line is $x \cos \alpha + y \sin \alpha = p$, where $p$ is the perpendicular distance from the origin. Substituting $\alpha = 45^{\circ}$, we have:

$x \cos 45^{\circ} + y \sin 45^{\circ} = p$

Since $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$ and $\sin 45^{\circ} = \frac{1}{\sqrt{2}}$, the equation becomes:

$\frac{x}{\sqrt{2}} + \frac{y}{\sqrt{2}} = p$

Multiplying both sides by $\sqrt{2}$ yields:

$x + y = p\sqrt{2}$

Rearranging into the general form:

$x + y - p\sqrt{2} = 0$

We compare this equation with the given equation of line L: $x + by + c = 0$. Since both equations represent the same line, their coefficients must be proportional:

$\frac{1}{1} = \frac{b}{1} = \frac{c}{-p\sqrt{2}}$

From the first equality, $\frac{1}{1} = \frac{b}{1}$, we deduce:

$b = 1$

From the second equality, $\frac{1}{1} = \frac{c}{-p\sqrt{2}}$, we get:

$c = -p\sqrt{2}$ Since $p$ is a positive distance ($p > 0$), $c$ must be negative. This is consistent.

Step 4: Calculate the value of $b^2 + c^2$.

We have found $b = 1$. Substituting $b = 1$ into equation $(*)$:

$\frac{c^2}{|1|} = 96$

$c^2 = 96$

Finally, we calculate $b^2 + c^2$:

$b^2 + c^2 = (1)^2 + 96 = 1 + 96 = 97$

Common Mistakes & Tips

• Sign Errors: Be extremely careful with signs, especially when dealing with intercepts and the normal form of the line. The perpendicular distance 'p' is always positive.
• Confusion with variables: Distinguish between the y-intercept 'b'' and the coefficient 'b' in the general form of the line equation.
• Absolute Values: Remember to use absolute values when calculating areas, as the area must be a positive quantity.

Summary

By utilizing the intercept form, the area formula, and the normal form of a straight line, we established relationships between the coefficients $b$ and $c$. We determined that $b=1$ and $c^2 = 96$. Consequently, we calculated $b^2+c^2$ to be 97.

The final answer is $\boxed{97}$, which corresponds to option (D).