Key Concepts and Formulas

• Equation of a Straight Line: $ax + by + c = 0$ represents a straight line.
• Intercepts: The x-intercept is found by setting $y=0$, and the y-intercept is found by setting $x=0$.
• Foot of the Perpendicular: If the foot of the perpendicular from $(x_1, y_1)$ to the line $ax + by + c = 0$ is $(h, k)$, then $\frac{h - x_1}{a} = \frac{k - y_1}{b} = -\frac{ax_1 + by_1 + c}{a^2 + b^2}$.
• Section Formula: If point P divides the line segment joining A($x_1, y_1$) and B($x_2, y_2$) in the ratio m:n, then the coordinates of P are given by $P = (\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n})$.

Step-by-Step Solution

Step 1: Find the coordinates of A and B.

The given line is $3x + y = \lambda$. To find the x-intercept (point A), set $y = 0$: $3x + 0 = \lambda \Rightarrow x = \frac{\lambda}{3}$. Thus, $A = (\frac{\lambda}{3}, 0)$. To find the y-intercept (point B), set $x = 0$: $3(0) + y = \lambda \Rightarrow y = \lambda$. Thus, $B = (0, \lambda)$.

Step 2: Find the coordinates of the foot of the perpendicular (P) from the origin to the line.

Let $P = (h, k)$. We use the formula for the foot of the perpendicular from $(0, 0)$ to the line $3x + y - \lambda = 0$: $\frac{h - 0}{3} = \frac{k - 0}{1} = -\frac{3(0) + 1(0) - \lambda}{3^2 + 1^2} = \frac{\lambda}{10}$ So, $h = \frac{3\lambda}{10}$ and $k = \frac{\lambda}{10}$. Thus, $P = (\frac{3\lambda}{10}, \frac{\lambda}{10})$.

Step 3: Find the ratio BP : PA using the section formula (or distance formula).

Let the ratio BP : PA be $m : n$. Then the coordinates of P can be expressed as: $P = (\frac{m(\frac{\lambda}{3}) + n(0)}{m+n}, \frac{m(0) + n(\lambda)}{m+n}) = (\frac{m\lambda}{3(m+n)}, \frac{n\lambda}{m+n})$ Comparing this with the coordinates of P we found earlier, $P = (\frac{3\lambda}{10}, \frac{\lambda}{10})$, we have: $\frac{m\lambda}{3(m+n)} = \frac{3\lambda}{10}$ and $\frac{n\lambda}{m+n} = \frac{\lambda}{10}$ From the second equation: $\frac{n}{m+n} = \frac{1}{10} \Rightarrow 10n = m+n \Rightarrow m = 9n$ Therefore, $\frac{m}{n} = \frac{9}{1}$. Thus, BP : PA = 9 : 1. However, the given answer is 1:3. Let's verify our work with the section formula again, and check if we made a mistake. Let's assume PA:BP = k:1. Then, P = $\frac{kA + B}{k+1}$. $(\frac{3\lambda}{10}, \frac{\lambda}{10}) = (\frac{k\frac{\lambda}{3} + 0}{k+1}, \frac{0 + \lambda}{k+1})$. $\frac{\lambda}{10} = \frac{\lambda}{k+1}$. $k+1 = 10 \implies k = 9$. $\frac{3\lambda}{10} = \frac{k\frac{\lambda}{3}}{k+1} = \frac{9\frac{\lambda}{3}}{10} = \frac{3\lambda}{10}$.

Since we got 9:1, there might be an error in the question/given answer. Let's re-examine the options and the problem. BP:PA = 9:1. This means PA is shorter than BP.

Let's consider PA:BP = 1:3. $P = \frac{1A + 3B}{1+3} = (\frac{\frac{\lambda}{3}}{4}, \frac{3\lambda}{4}) = (\frac{\lambda}{12}, \frac{3\lambda}{4})$. This does not equal the P calculated earlier.

We made an error in our ratio calculation. We have A = ($\frac{\lambda}{3}$, 0) and B = (0, $\lambda$) and P = ($\frac{3\lambda}{10}$, $\frac{\lambda}{10}$). Let BP:PA = r:1. Then P = $\frac{rA + B}{r+1}$. ($\frac{3\lambda}{10}$, $\frac{\lambda}{10}$) = ($\frac{r\frac{\lambda}{3} + 0}{r+1}$, $\frac{0 + \lambda}{r+1}$). $\frac{\lambda}{10} = \frac{\lambda}{r+1}$. Thus $r+1 = 10$, so $r=9$. BP:PA = 9:1. Let PA:BP = r:1. Then P = $\frac{rB+A}{r+1} = (\frac{\lambda/3}{r+1}, \frac{r\lambda}{r+1})$. $\frac{3\lambda}{10} = \frac{\lambda/3}{r+1}$. $\frac{\lambda}{10} = \frac{r\lambda}{r+1}$. $3(r+1) = 10/3 \implies r+1 = 10/9$. $r+1 = 10r \implies 9r = 1 \implies r = 1/9$. Then PA:BP = 1/9. So BP:PA = 9:1.

There seems to be an error in the given answer. After reviewing the solution, the ratio BP:PA is 9:1.

Common Mistakes & Tips

• Carefully apply the section formula to avoid errors in calculating the ratio.
• Double-check the coordinates of points A, B, and P before calculating the ratio.
• Be mindful of the order of the ratio (BP:PA vs. PA:BP).

Summary

We found the coordinates of the x and y intercepts (A and B) of the given line. Then, we found the coordinates of the foot of the perpendicular (P) from the origin to the line. Finally, we used the section formula to determine the ratio in which P divides the line segment AB. The ratio BP:PA is calculated to be 9:1.

Final Answer

The final answer is \boxed{9:1}, which corresponds to option (D).