If the distance of the point P ( 43 ,0 , from the line r = 4 i - k + ( 2 i + 3 k ) , along a line with direction ratios 3 , - 1 , 0 is 13 10 , then 2 + 2 is equal to _______ [2026]

Question

If the distance of the point $P (43, α, β), β < 0,$ from the line $\vec{r} = 4 \hat{i} - \hat{k} + μ (2 \hat{i} + 3 \hat{k}), μ \in ℝ$ along a line with direction ratios $3, - 1, 0$ is $13 \sqrt{10},$ then $α^{2} + β^{2}$ is equal to _______ [2026]

Smart Achievers · Accepted Answer

(170)

$\frac{x - 43}{3} = \frac{y - α}{- 1} = \frac{z - β}{0} \Rightarrow P_{1} (43 + 3 λ, α - λ, β)$

$\frac{x - 4}{2} = \frac{y}{0} = \frac{z + 1}{3} \Rightarrow P_{1} (2 μ + 4, 0, 3 μ - 1)$

$∴$ $μ = \frac{3 λ + 39}{2}, α = λ, β = \frac{9 λ - 115}{2}$

$P (43, α, β), P_{1} (43 + 3 α, 0, β)$

${(P P_{1})}^{2} = 1690 = 10 α^{2} \Rightarrow α = 13, β = 1$

$∴$ $α^{2} + β^{2} = 170$

Solution

Q.
If the distance of the point $P (43, α, β), β < 0,$ from the line $\vec{r} = 4 \hat{i} - \hat{k} + μ (2 \hat{i} + 3 \hat{k}), μ \in ℝ$ along a line with direction ratios $3, - 1, 0$ is $13 \sqrt{10},$ then $α^{2} + β^{2}$ is equal to _______ [2026]

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Solution

Q. If the distance of the point P(43,α,β), β<0, from the line r→=4i^-k^+μ(2i^+3k^), μ∈ℝ along a line with direction ratios 3, -1, 0 is 1310, then α2+β2 is equal to _______ [2026]

Ans. (170) x-433=y-α-1=z-β0⇒P1(43+3λ, α-λ, β) x-42=y0=z+13⇒P1(2μ+4, 0, 3μ-1) ∴ μ=3λ+392, α=λ, β=9λ-1152 P(43, α, β), P1(43+3α, 0, β) (PP1)2=1690=10α2⇒α=13, β=1 ∴ α2+β2=170

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Q.
If the distance of the point $P (43, α, β), β < 0,$ from the line $\vec{r} = 4 \hat{i} - \hat{k} + μ (2 \hat{i} + 3 \hat{k}), μ \in ℝ$ along a line with direction ratios $3, - 1, 0$ is $13 \sqrt{10},$ then $α^{2} + β^{2}$ is equal to _______ [2026]