If the line (x-1)/1=(y-k)/(-2)=(z-3)/(λ) lies in the plane 3x+4y-2z=6, then 5k+3λ is equal to

Correct answer: (B) 75/4. If the line (x-1)/1=(y-k)/(-2)=(z-3)/(λ) lies in the plane 3x+4y-2z=6 ⇒1,k,3 satisfies the equation of the plane ⇒3+4k-6=6⇒k=9/4 And b → ⋅ n → =0 b → = 1,−2,λ and n → = 3,4,−2 ⇒3-8-2λ=0 ⇒λ=-5/2 5k+3λ=45/4+15/2=75/4

Q42 / 105 · NTA Abhyas (JEE Main) 2020 · Mathematics · Three Dimensional Geometry

If the line $\frac{x - 1}{1} = \frac{y - k}{- 2} = \frac{z - 3}{λ}$ lies in the plane $3 x + 4 y - 2 z = 6,$ then $5 |k| + 3 |λ|$ is equal to

A
$75$
Correct answer
B
$\frac{75}{4}$
C
$15$
D
$\frac{5}{2}$

Solution

If the line

\frac{x - 1}{1} = \frac{y - k}{- 2} = \frac{z - 3}{λ}

lies in the plane

3 x + 4 y - 2 z = 6

\Rightarrow (1, k, 3)

satisfies the equation of the plane

\Rightarrow 3 + 4 k - 6 = 6 \Rightarrow k = \frac{9}{4}

And

\vec{b} \cdot \vec{n} = 0 (\vec{b} = 〈1, - 2, λ〉 and \vec{n} = 〈3, 4, - 2〉)

\Rightarrow 3 - 8 - 2 λ = 0

\Rightarrow λ = - \frac{5}{2}

5 |k| + 3 |λ| = \frac{45}{4} + \frac{15}{2} = \frac{75}{4}

Hence, the correct answer is option: 'B'.

If the line x-11=y-k-2=z-3λ lies in the plane 3x+4y-2z=6, then 5k+3λ is equal to

Solution

More Three Dimensional Geometry Questions

If the line $\frac{x - 1}{1} = \frac{y - k}{- 2} = \frac{z - 3}{λ}$ lies in the plane $3 x + 4 y - 2 z = 6,$ then $5 |k| + 3 |λ|$ is equal to