JEE Main 2022 · 25 Jun · Shift 1 · Mathematics Q28

← JEE Main 2022 · 25 Jun · Shift 1

MathematicsInteger+4 / -1Question 28 of 90

Let the lines
$L_1:\vec r=\lambda\left(\hat i+2\hat j+3\hat k\right),\ \lambda\in\mathbb R$
$L_2:\vec r=\left(\hat i+3\hat j+\hat k\right)+\mu\left(\hat i+\hat j+5\hat k\right);\ \mu\in\mathbb R,$
intersect at the point S. If a plane $ax+by-z+d=0$ passes through S and is parallel to both the lines $L_1$ and $L_2$ , then the value of $a+b+d$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 5

Intersection. $\lambda=1+\mu$ , $2\lambda=3+\mu$ give $\mu=1$ , $\lambda=2$ (the $z$ -components agree): $S(2,4,6)$ .

Normal. $(1,2,3)\times(1,1,5)=(7,-2,-1)$ , so $a=7$ , $b=-2$ .

Through S. $14-8-6+d=0$ , $d=0$ . So $a+b+d=5$ .

Answer: 5

Topics:Equation of a plane Equation of a line in space Cross product

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JEE Main 2022 · 25 Jun · Shift 1, Mathematics question 28