Question

Select the correct option from the given options.

Answer 1

Dear Student:

Given:

 r=(2,-3,7)+k(2,a,5),   k∈R

And other r = (1,2,3)+k(3,-a,a),     k∈R

Here,(2,a,5) and (3,-a,a) are direction ration of given vector.

So if both are perpendicular to each other.

Then dot product of direction ratio will be zero.

$(2,a,5).(3,-a,a)=0$

$6-a^{2}+5a=0$

$a^{2}-5a-6=0$

$a^{2}-6a+a-6=0$

$a(a-6)+1(a-6)=0$

$(a-6)(a+1)=0$

$(a-6)=0$

$a=6$

$(a+1)=0$

$a=-1$

So,option d is correct.

Answer 2

Given two lines $\overline{r}=(2,-3,7)+k(2,a,5),k\in R$ and $\overline{r}=(1,2,3)+k(3,-a,a),k\in R$ are perpendicular to each other.
therefore, dot product of direction ratios of given lines.
so, dot product of (2, a,5) and (3,-a,a) = 0

e.g., $(2,a,5).(3,-a,a)$ = 0

or, $(2\hat{i}+a\hat{j}+5\hat{k}).(3\hat{i}-a\hat{j}+a\hat{k})=0$

or, 2 × 3 + a × -a + 5 × a = 0

or, 6 - a² + 5a = 0

or, a² - 5a - 6 = 0

or, a² - 6a + a - 6 = 0

or, a(a - 6) + 1(a - 6) = 0

or, (a + 1)(a - 6) = 0

or, a = -1 and 6

here, option (d) is matched with value of a.
so, option (d) is correct choice