We will cover following topics

Straight Lines
Shortest Distance Between Two Skew Lines
PYQs
- Straight Lines
- Shortest Distance Between Two Skew Lines

Straight Lines

General equation of the line

The general equation of a straight line in the cartesian coordinate system is given by $A x + B y + C = 0$ , where the vector $(A, B)$ is the normal vector to the straight line.

Slope-Intercept form of the line

The slope-intercept form of the line is given by

y = m x + c

where $m$ is the slope and $c$ is the $y - i n t e r c e p t$ .

Equation of a line passing through two points

The equation of a straight line passing through two-points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is given by

$\frac{y - y_{1}}{y_{2} - y_{1}} = \frac{x - x_{1}}{x_{2} - x_{1}}$ or $| \begin{matrix} x & y & 1 \\ x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \end{matrix} | = 0$

Intercept form of the Line

The intercept form of a straight line is given by

\frac{x}{a} + \frac{y}{b} = 1

where $a$ and $b$ are $x - i n t e r c e p t$ and $y - i n t e r c e p t$ respectively.

Vector equation of the Line

The vector equation of the line is given by

\vec{r} = \vec{a} + t \vec{b}

where $\vec{a}$ lies on the line and $\vec{b}$ determines the direction of the line.

Point direction form of the Line

The point direction form of the line is given by

\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}

where $P_{1} (x_{1}, y_{1}, z_{1})$ lies on the line and $s = (a, b, c)$ is the direction vector of the line.

Distance from a Point to a Line

The distance $d$ from a point $M (x_{1}, y_{1})$ to the line $A x + B y + C = 0$ is given by:

d = \frac{| A x_{1} + B y_{1} + C |}{\sqrt{A^{2} + B^{2}}}

Condition for two lines to be Parallel

Two straight lines $y = m_{1} x + c_{1}$ and $y = m_{2} x + c_{2}$ are parallel if $k_{1} = k_{2}$ .
Similarly, $A_{1} x + B_{1} y + C_{1} = 0$ and $A_{2} x + B_{2} y + C_{2} = 0$ are parallel if $\frac{A_{1}}{A_{2}} = \frac{B_{1}}{B_{2}}$ .
Also, if the lines are represented in point direction forms, then they are parallel if their direction vectors $s_{1} (a_{1}, b_{1}, c_{1})$ and $s_{2} (a_{2}, b_{2}, c_{2})$ are collinear, i.e., $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ .

Condition for two lines to be Perpendicular

Two straight lines $y = m_{1} x + c_{1}$ and $y = m_{2} x + c_{2}$ are perpendicular if $k_{1} k_{2} = - 1$ . Similarly, $A_{1} x + B_{1} y + C_{1} = 0$ and $A_{2} x + B_{2} y + C_{2} = 0$ are perpendicular if $A_{1} A_{2} + B_{1} B_{2} = 0$ .
Also, if the lines are represented in point direction forms, then they are perpendicular if the dot product of their direction vectors is 0, i.e., $s_{1} \cdot s_{2} = 0$ , or $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$ .

Angle between two lines

The angle between two lines is given by:

\tan φ = \frac{m_{2} - m_{1}}{1 + m_{1} m_{2}}

when lines are represented in slope-intercept forms,

\cos φ = \frac{A_{1} A_{2} + B_{1} B_{2}}{\sqrt{A_{1}^{2} + B_{1}^{2}} \sqrt{A_{2}^{2} + B_{2}^{2}}}

when lines are represented by their general equations, and $\cos φ = \frac{s_{1} \cdot s_{2}}{| s_{1} | \cdot | s_{2} |} = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \cdot \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}}$

when the lines are represented in their point direction forms.

Condition of Interesection of two lines

Two lines $\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}$ and $\frac{x - x_{2}}{a_{2}} = \frac{y - y_{2}}{b_{2}} = \frac{z - z_{2}}{c_{2}}$ intersect if:

| \begin{matrix} x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{matrix} |

Intersection point of two lines

The intersection point of two straight lines $A_{1} x + B_{1} y + C_{1} = 0$ and $A_{2} x + B_{2} y + C_{2} = 0$ is given by:

x_{0} = \frac{- C_{1} B_{2} + C_{2} B_{1}}{A_{1} B_{2} - A_{2} B_{1}}

y_{0} = \frac{- A_{1} C_{2} + A_{2} C_{1}}{A_{1} B_{2} - A_{2} B_{1}}

The straight line given by $\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$ and the plane given by $A x + B y + C z + D = 0$ are:
(i) Parallel, if $n \cdot s = 0$ , or $A a + B b + C c = 0$
(ii) Perpendicular, if $n ‖ s$ or $\frac{A}{a} = \frac{B}{b} = \frac{C}{c}$

Shortest Distance Between Two Skew Lines

The shortest distance between two skew lines is the length of the mutual perpendicular between the two lines.

Let the two lines be given by: ${\vec{r}}_{1} = {\vec{a}}_{1} + t \cdot {\vec{b}}_{1}$

{\vec{r}}_{2} = {\vec{a}}_{2} + t \cdot {\vec{b}}_{2}

where $P = {\vec{a}}_{1}$ lies on $l_{1}$ and $Q = {\vec{a}}_{2}$ lies on $l_{2}$ .

The unit vector normal to both $l_{1}$ and $l_{2}$ is given by $\frac{({\vec{b}}_{1} \times {\vec{b}}_{2})}{| {\vec{b}}_{1} \times {\vec{b}}_{2} |}$ .

Now, the projection of $P Q$ on the normal will be the shortest distance between $l_{1}$ and $l_{2}$ , and is given by:

\frac{| ({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) |}{| {\vec{b}}_{1} \times {\vec{b}}_{2} |}

PYQs

Straight Lines

1) Show that the lines $\frac{x + 1}{- 3} = \frac{y - 3}{2} = \frac{z + 2}{1}$ and $\frac{x}{1} = \frac{y - 7}{- 3} = \frac{z + 7}{2}$ intersect. Find the coordinates of the point of intersection and the equation of the plane containing them.

[10M]

2) Find the shortest distance between the lines

a_{1} x + b_{1} y + c_{1} z + d_{1} = 0

a_{2} x + b_{2} y + c_{2} z + d_{2} = 0

and the $z - a x i s$ .

[2018, 12M]

3) Find the surface generated by a line which intersects the line $y = a = z, x + 3 z = a = y + z$ and parallel to the plane $x + y = 0$ .

[2016, 10M]

4) Prove that two of the straight lines represented by the equation $x^{3} + b x^{2} y + c x y^{2} + y^{3} = 0$ will be at right angles, if $b + c = - 2$ .

[2012, 12M]

5) Find the equation of the straight line through the point (3,1,2) to intersect the straight line $x + 4 = y + 1 = 2 (z - 2)$ and parallel to the plane $4 x + y + 5 z = 0$ .

[2011, 10M]

6) A line with direction ratios 2,7,-5 is drawn to intersect the lines $\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{4}$ and $\frac{x - 11}{3} = \frac{y - 5}{- 1} = \frac{z}{1}$ . Find the coordinate of the points of intersection and the length intercepted on it.

[2007, 15M]

7) Prove that the locus of a line which meets the lines $y = \pm m x, z = \pm c$ and the circle $x^{2} + y^{2} = a^{2}$ , $z = 0$ is $c^{2} m^{2} (c y - m z x)^{2} + c^{2} (y z - c m x)^{2}$ = $a^{2} m^{2} {(z - c^{2})}^{2}$ .

[2004, 15M]

8) Find the equation of the two straight lines through the point (1,1,1) that intersect the line $x - 4 = 4 (y - 4) = 2 (z - 1)$ at an angle of $60^{\circ}$ .

[2003, 12M]

Shortest Distance Between Two Skew Lines

1) Find the shortest distance between the skew the lines:

\frac{x - 3}{3} = \frac{8 - y}{1} = \frac{z - 3}{1}

and

\frac{x + 3}{- 3} = \frac{y + 7}{2} = \frac{z - 6}{4}

[2017, 10M]

2) Find the shortest distance between the lines $\frac{x - 1}{2} = \frac{y - 2}{4} = z - 3$ and $y - m x = z = 0$ . For what value of $m$ will the two lines intersect?

[2016, 10M]

3) Show that the length of the shortest distance between the line $z = x \tan α, y = 0$ and any tangent to the ellipse $x^{2} \sin^{2} α + y^{2} = a^{2}, z = 0$ is constant.

[2006, 12M]

4) Find the equations of the lines of shortest distance between the lines: $y + z = 1$ , $x = 0$ and $x + z = 1, y = 0$ as the intersection of two planes.

[2003, 15M]

5) Find the shortest distance between the axis of $z$ and the lines $a x + b y + c z + d = 0$ , $a^{1} x + b^{1} y + c^{1} z + d^{1} = 0$ .

[2001, 15M]

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