Let and , . Let the projection of the vector on the diagonal of the parallelogram ABCD be of length one unit. If , where , be the roots of the equation then is equal to [2026]

STUDY MATERIALS
COURSES
MORE
SUCCESS STORY
ADMISSION
STORE

Back

JEE ADVANCE

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

JEE MAIN

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

NEET

CLASS XI - XII
CRASH COURSE

CLASS XI - XII

CRASH COURSE

BITSAT

CLASS XI - XII

CLASS XI - XII

CBSE BOARD

CLASS IX

CLASS X

CLASS XI

CLASS XII

CLASS VI

CLASS VII

CLASS VIII

UP BOARD

CLASS IX
CLASS X
CLASS XI
CLASS XII

CLASS IX

CLASS X

CLASS XI

CLASS XII

CUET

CRASH COURSE

CRASH COURSE

Courses

CLASSROOM COURSES
ONLINE COURSES

ONLINE COURSES

More

Solution

Q.
Let $\vec{A B} = 2 \hat{i} + 4 \hat{j} - 5 \hat{k}$ and $\vec{A D} = \hat{i} + 2 \hat{j} + λ \hat{k}$ , $λ \in ℝ$ . Let the projection of the vector $\vec{v} = \hat{i} + \hat{j} + \hat{k}$ on the diagonal $\vec{A C}$ of the parallelogram ABCD be of length one unit. If $α, β$ , where $α > β$ , be the roots of the equation $λ^{2} x^{2} - 6 λ x + 5 = 0,$ then $2 α - β$ is equal to [2026]

1 1

2 4

3 3

4 6

Ans.
(3)

$\vec{A C} = 3 \hat{i} + 6 \hat{j} + (λ - 5) \hat{k}$

$\vec{v} \cdot \vec{A C} = 1$ $\Rightarrow 3 + 6 + λ - 5 = \sqrt{9 + 36 + {(λ - 5)}^{2}}$

$\Rightarrow λ^{2} + 8 λ + 16 = λ^{2} - 10 λ + 70$

$\Rightarrow λ = \frac{54}{18} = 3$

$∴$ $Quadratic: 9 x^{2} - 18 x + 5 = 0$ $\Rightarrow x = \frac{1}{3}, \frac{5}{3}$

$∴$ $2 α - β = \frac{10 - 1}{3} = 3$

Want Us to Email you About Special Offers & Updates?