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Step-by-Step Solution
Step 1: Identify the given line
The problem provides the line 3x + y = λ, where λ ≠ 0. We need the foot of the perpendicular from the origin (0,0) onto this line. We also need the points where the line intersects the x-axis and y-axis, and then find the ratio BP : PA.
Step 2: Write the equation of the perpendicular from the origin
The slope of the line 3x + y = λ is found by rewriting it in slope-intercept form:
3x + y = λ
⇒ y = −3x + λ.
Thus, its slope is −3.
A line perpendicular to this line has slope mperp = 1/3.
Since the perpendicular passes through the origin (0,0), its equation can be written as y = (1/3)x or, equivalently,
3y − x = 0.
Another way (as shown in the solution outline) is to use the fact that the directional ratios for 3x + y = λ are (3,1). A perpendicular direction ratio to (3,1) is (1,−3), but we typically find the simpler slope form. Both approaches are valid.
Step 3: Determine the foot of the perpendicular (point P)
The foot of the perpendicular, P, is the point of intersection between the lines 3x + y = λ and y = (1/3)x.
Let P = (xP, yP). Substituting y = (1/3)x into 3x + y = λ gives:
3x + (1/3)x = λ
⇒ (10/3)x = λ
⇒ x = (3λ/10).
Since y = (1/3)x, we have y = (1/3)(3λ/10) = λ/10.
So, P = (3λ/10, λ/10).
Step 4: Find the intersection with the x-axis (point A)
On the x-axis, y = 0. Substitute y = 0 into 3x + y = λ:
3x + 0 = λ
⇒ x = λ/3.
Hence, A = (λ/3, 0).
Step 5: Find the intersection with the y-axis (point B)
On the y-axis, x = 0. Substitute x = 0 into 3x + y = λ:
3(0) + y = λ
⇒ y = λ.
Hence, B = (0, λ).
Step 6: Compute the lengths BP and PA
BP: The coordinates of B are (0, λ), and the coordinates of P are (3λ/10, λ/10). The distance BP is:
$BP = \sqrt{(0 - \frac{3\lambda}{10})^2 + (\lambda - \frac{\lambda}{10})^2} \\
= \sqrt{\left(-\frac{3\lambda}{10}\right)^2 + \left(\frac{9\lambda}{10}\right)^2} \\
= \sqrt{\frac{9\lambda^2}{100} + \frac{81\lambda^2}{100}} \\
= \sqrt{\frac{90\lambda^2}{100}} \\
= \sqrt{\frac{90}{100}}\,|\lambda| \\
= \frac{\sqrt{90}}{10}\,|\lambda|\text{.}$
PA: The coordinates of A are $(\frac{\lambda}{3}, 0)$, and the coordinates of P are $(\frac{3\lambda}{10}, \frac{\lambda}{10})$. Then,
$PA = \sqrt{\left(\frac{\lambda}{3} - \frac{3\lambda}{10}\right)^2 + \left(0 - \frac{\lambda}{10}\right)^2}.$
First, compute each difference:
$\frac{\lambda}{3} - \frac{3\lambda}{10} = \frac{10\lambda - 9\lambda}{30} = \frac{\lambda}{30},$
and
$0 - \frac{\lambda}{10} = - \frac{\lambda}{10}.$
Thus,
$PA = \sqrt{\left(\frac{\lambda}{30}\right)^2 + \left(-\frac{\lambda}{10}\right)^2}
= \sqrt{\frac{\lambda^2}{900} + \frac{\lambda^2}{100}}
= \sqrt{\frac{\lambda^2}{900} + \frac{9\lambda^2}{900}}
= \sqrt{\frac{10\lambda^2}{900}}
= \sqrt{\frac{10}{900}}\,|\lambda|
= \frac{\sqrt{10}}{30}\,|\lambda|.$
Step 7: Find the ratio BP : PA
From the above expressions:
$BP = \frac{\sqrt{90}}{10}\,|\lambda| = \frac{3\sqrt{10}}{10}\,|\lambda|,$
and
$PA = \frac{\sqrt{10}}{30}\,|\lambda|.$
Therefore, the ratio is:
$\frac{BP}{PA} = \frac{\frac{3\sqrt{10}}{10} \,|\lambda|}{\frac{\sqrt{10}}{30}\,|\lambda|} = \frac{3\sqrt{10}}{10} \times \frac{30}{\sqrt{10}} = 3 \times 3 = 9.$
Hence,
$BP : PA = 9 : 1.$
Final Answer: 9 : 1
