Lines And Angles

The parametric equations of the lines

L_1

and

L_2

are given as follows: For line

L_1

, passing through

(7%2C%206%2C%202)

and parallel to the vector

%5Cmathbf%7Ba%7D%20%3D%20-3%5Chat%7Bi%7D%20%2B%202%5Chat%7Bj%7D%20%2B%204%5Chat%7Bk%7D

L_1%3A%20(7%20%2B%20%5Clambda(-3)%2C%206%20%2B%20%5Clambda(2)%2C%202%20%2B%20%5Clambda(4))%20%5Cquad%20%5Ctext%7B(Equation%20of%20line%201)%7D

For line

L_2

, passing through

(5%2C%203%2C%204)

and parallel to the vector

%5Cmathbf%7Bb%7D%20%3D%202%5Chat%7Bi%7D%20%2B%20%5Chat%7Bj%7D%20%2B%203%5Chat%7Bk%7D

L_2%3A%20(5%20%2B%20%5Clambda(3)%2C%203%20%2B%20%5Clambda(1)%2C%204%20%2B%20%5Clambda(3))%20%5Cquad%20%5Ctext%7B(Equation%20of%20line%202)%7D

The shortest distance between skew lines is given by the formula:

d%20%3D%20%5Cfrac%7B%7C%20(%5Cmathbf%7Bb%7D_1%20-%20%5Cmathbf%7Bb%7D_2)%20%5Ccdot%20(%5Cmathbf%7Ba%7D_1%20%5Ctimes%20%5Cmathbf%7Ba%7D_2)%20%7C%7D%7B%7C%5Cmathbf%7Ba%7D_1%20%5Ctimes%20%5Cmathbf%7Ba%7D_2%7C%7D

Where

%5Cmathbf%7Ba%7D_1

and

%5Cmathbf%7Ba%7D_2

are the direction vectors of the two lines, and

%5Cmathbf%7Bb%7D_1

and

%5Cmathbf%7Bb%7D_2

are points on the respective lines. Here,

%5Cmathbf%7Ba%7D_1%20%3D%20(-3%2C%202%2C%204)

%5Cmathbf%7Ba%7D_2%20%3D%20(2%2C%201%2C%203)

%5Cmathbf%7Bb%7D_1%20%3D%20(7%2C%206%2C%202)

, and

%5Cmathbf%7Bb%7D_2%20%3D%20(5%2C%203%2C%204)

. We compute the cross product

%5Cmathbf%7Ba%7D_1%20%5Ctimes%20%5Cmathbf%7Ba%7D_2

first:

%5Cmathbf%7Ba%7D_1%20%5Ctimes%20%5Cmathbf%7Ba%7D_2%20%3D%20%5Cbegin%7Bvmatrix%7D%20%5Chat%7Bi%7D%20%26%20%5Chat%7Bj%7D%20%26%20%5Chat%7Bk%7D%20-3%20%26%202%20%26%204%202%20%26%201%20%26%203%20%5Cend%7Bvmatrix%7D%0A%3D%20(2%20%5Ctimes%203%20-%204%20%5Ctimes%201)%20%5Chat%7Bi%7D%20-%20(-3%20%5Ctimes%203%20-%204%20%5Ctimes%202)%20%5Chat%7Bj%7D%20%2B%20(-3%20%5Ctimes%201%20-%202%20%5Ctimes%202)%20%5Chat%7Bk%7D

%3D%20(6%20-%204)%5Chat%7Bi%7D%20-%20(-9%20-%208)%5Chat%7Bj%7D%20%2B%20(-3%20-%204)%5Chat%7Bk%7D

%3D%202%20%5Chat%7Bi%7D%20%2B%2017%20%5Chat%7Bj%7D%20-%207%20%5Chat%7Bk%7D

Now, the distance is calculated using the formula above:

d%20%3D%20%5Cfrac%7B%7C(2%20%5Chat%7Bi%7D%20%2B%2017%20%5Chat%7Bj%7D%20-%207%20%5Chat%7Bk%7D)%20%5Ccdot%20(2%20%5Chat%7Bi%7D%20%2B%2017%20%5Chat%7Bj%7D%20-%207%20%5Chat%7Bk%7D)%7C%7D%7B%5Csqrt%7B342%7D%7D%20%3D%20%5Cfrac%7B69%7D%7B%5Csqrt%7B342%7D%7D%20%3D%20%5Cfrac%7B23%7D%7B%5Csqrt%7B38%7D%7D

Thus, the shortest distance between the two lines is

%5Cfrac%7B23%7D%7B%5Csqrt%7B38%7D%7D

The line $L$ passes through $(1%2C%201%2C%201)$ , so the parametric equations for line $L$ can be written as: $x%20%3D%201%20%2B%202t%2C%20%5Cquad%20y%20%3D%201%20%2B%203t%2C%20%5Cquad%20z%20%3D%201%20%2B%204t.$ Now, consider the first line: $%5Cfrac%7Bx%20-%201%7D%7B2%7D%20%3D%20%5Cfrac%7By%20%2B%201%7D%7B3%7D%20%3D%20%5Cfrac%7Bz%20-%201%7D%7B4%7D.$ Let the common ratio for this line be $k$ . So, we can write the parametric equations for this line as: $x%20%3D%201%20%2B%202k%2C%20%5Cquad%20y%20%3D%20-1%20%2B%203k%2C%20%5Cquad%20z%20%3D%201%20%2B%204k.$ Now, consider the second line: $%5Cfrac%7Bx%20-%203%7D%7B1%7D%20%3D%20%5Cfrac%7By%20-%204%7D%7B2%7D%20%3D%20%5Cfrac%7Bz%7D%7B1%7D.$ Let the common ratio for this line be $m$ . So, we can write the parametric equations for this line as: $x%20%3D%203%20%2B%20m%2C%20%5Cquad%20y%20%3D%204%20%2B%202m%2C%20%5Cquad%20z%20%3D%20m.$ We now need to find the value of $t$ where line $L$ intersects the two given lines.
The parametric equations of line $L$ and the first line give us a system of equations.
Similarly, the parametric equations of line $L$ and the second line also give us another system of equations.
Solving these equations yields the value of $t$ , and the corresponding point on line $L$ .
After solving the system, we find that the point $(7%2C%2015%2C%2013)$ lies on the line $L$ , as this point satisfies both the intersection conditions.
Thus, the correct answer is: $(7%2C%2015%2C%2013).$

About Lines And Angles - JEE-MAIN

Lines And Angles is a vital chapter for JEE-MAIN aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Lines And Angles PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Lines And Angles carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

About Lines And Angles - JEE-MAIN

Frequently Asked Questions

Why focus on Lines And Angles PYQs?

How to best use this analysis?

Lines And Angles

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

About Lines And Angles - JEE-MAIN

Frequently Asked Questions

Why focus on Lines And Angles PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs