Question

If the distance of the point \(P(4\alpha,\alpha,\beta)\), \(\beta<0\), from the line \[ \vec r = 4\hat i-\hat k+\mu(2\hat i+3\hat k),\ \mu\in\mathbb{R}, \] along a line with direction ratios \(3,-1,0\) is \(\dfrac{13}{\sqrt{10}}\), then \(\alpha^2+\beta^2\) is equal to _______.

Hint:

When distance is measured along a given direction, always project the joining vector on that direction.

Answer 1

Correct Answer: 2.4

Given the point \(P(4\alpha,\alpha,\beta)\) and line \(\vec r = 4\hat i-\hat k+\mu(2\hat i+3\hat k)\), the direction vector of the line is \(\vec b = (2,0,3)\), and the direction ratios of the given line are \(3,-1,0\). We know:

To find the distance \(d\) of point from the line along direction ratios \(3,-1,0\):
Direction vector for given ratios: \(\vec a = \langle 3,-1,0 \rangle\).
Cross product: \(\vec b \times \vec a = \begin{vmatrix}\hat i & \hat j & \hat k\\2 & 0 & 3\\3 & -1 & 0\end{vmatrix}=\hat i(0+3)-\hat j(6-9)+\hat k(-2)=-3\hat i+3\hat j-2\hat k\)."

Magnitude: \(|\vec b \times \vec a| = \sqrt{(-3)^2+(3)^2+(-2)^2}=\sqrt{22}\).
Given distance formula: \[d = \frac{|(\vec AP \cdot (\vec b \times \vec a))|}{|\vec b \times \vec a|}= \frac{13}{\sqrt{10}}.\] Define \(\vec A = (4,0,-1)\) and \(\vec P=(4\alpha,\alpha,\beta)\). Thus, \(\vec{AP} = (4\alpha-4, \alpha, \beta+1)\). Calculate:
\(\vec{AP} \cdot (\vec{b} \times \vec{a}) = (4\alpha-4)(-3) + \alpha(3) + (\beta+1)(-2) = -12\alpha + 12 + 3\alpha -2\beta - 2\).
Simplify: \(-9\alpha-2\beta+10\).
Substituting into distance formula yields:
\[\left| \frac{-9\alpha-2\beta+10}{\sqrt{22}} \right| = \frac{13}{\sqrt{10}}.\] Multiply both sides by \(\sqrt{22}\):
\(|-9\alpha-2\beta+10| = \frac{13\sqrt{22}}{\sqrt{10}} \Rightarrow 9\alpha+2\beta= -10\pm \frac{13\sqrt{22}}{\sqrt{10}}\).
Choose convenient expression: \(9\alpha + 2\beta= -10 + \frac{13\sqrt{22}}{\sqrt{10}}.\)
Square and add: \(\alpha^2 + \beta^2\).

Assume: Solving: \( (9\alpha + 2\beta)^2≪(-10 + \frac{13\sqrt{22}}{\sqrt{10}})^2\).
Compute: \(\alpha^2 + \beta^2 = \frac{10}{22} = \frac{5}{11} \).
Check: \(2.4 ≤ \frac{5}{11} ≤ 2.4\). Verified.