Topics Covered

1. Distance of a line from a point

2. Distance between two parallel lines

3. Area of a parallelogram

4. Area of triangle

1. Distance of a line from a point

Length of perpendicular (distance) from the point $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ to the line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$ is given by

$\mathrm{p}=\frac{\left|\mathrm{ax} _{1}+\mathrm{by} _{1}+\mathrm{c}\right|}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}$

Length of perpendicular from the origin to the line $a x+b y+c=0$ is given by

$\mathrm{p}=\frac{|\mathrm{c}|}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}},\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right)$ is $(0,0)$

2. Distance between two parallel lines

The distance between two parallel lines $a x+b y+c _{1}=0$ and $a x+b y+c _{2}=0$ is given by

$\mathrm{p}=\frac{\left|\mathrm{c}_1-\mathrm{c}_2\right|}{\sqrt{\mathrm{a}^2+\mathrm{b}^2}}$

Note that coefficient of $x$ and $y$ of both equation must be same.

3. Area of a parallelogram

Area of a parallelogram $\mathrm{ABCD}=2$ area of $\triangle \mathrm{ABD}$

$ \begin{aligned} & \mathrm{A}=\mathrm{z}\left(\frac{1}{2} \times \mathrm{AB} \times \mathrm{AD} \times \sin \theta\right) \\ & \mathrm{A}=\mathrm{AB} \times \mathrm{AD} \times \sin \theta \\ & \text { Here }=\frac{\mathrm{PL}}{\mathrm{AD}}=\sin \theta \text { and } \frac{\mathrm{P} 2}{\mathrm{AD}}=\sin \theta \\ & A=\left(\frac{\mathrm{P} _{2}}{\sin \theta}\right)\left(\frac{\mathrm{P} _{1}}{\sin \theta}\right) x \sin \theta \\ & \mathrm{A}=\frac{p _{1} p _{2}}{\sin \theta} \end{aligned} $

If the sides of a parallelogram be $a _{1} x+b _{1} y+c _{1}=0 ; a _{1} x+b _{1} y+d _{1}=0: a _{2} x+b _{2} y+c _{2}=0, a _{2} x+b _{2} y+d _{2}=$ 0 area of parallelogram $=\left|\frac{\left(c _{1}-d _{1}\right)\left(c _{2}-d _{2}\right)}{\left|\begin{array}{ll}a _{1} & b _{1} \ a _{2} & b _{2}\end{array}\right|}\right|$

Area of Rhombus

In case of a rhombus $\mathrm{p} _{1}=\mathrm{p} _{2}$, area of rhombus $=\frac{p _{1}^{2}}{\sin \theta}$

Also area of rhombus $=\frac{1}{2} \mathrm{~d} _{1} \mathrm{~d} _{2}$ where $\mathrm{d} _{1}, \mathrm{~d} _{2}$ are the lengths of two perpendicular diagonals

4. Area of a triangle

Area of a triangle whose vertices are $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right),\left(\mathrm{x} _{2}, \mathrm{y} _{2}\right)$ and $\left(\mathrm{x} _{3}, \mathrm{y} _{3}\right)$ is

$ \frac{1}{2}\left|x _{1}\left(y _{2}-y _{3}\right)+x _{2}\left(y _{3}-y _{1}\right)+x _{3}\left(y _{2}-y _{1}\right)\right| $

$ \begin{aligned} & \text { Or } \frac{1}{2}\left|\begin{array}{ll} \mathrm{x} _{1} & \mathrm{y} _{1} \\ \mathrm{x} _{2} & \mathrm{y} _{2} \\ \mathrm{x} _{3} & \mathrm{y} _{3} \\ \mathrm{x} _{1} & \mathrm{y} _{1} \end{array}\right| \\ & =\frac{1}{2}\left[\left(\mathrm{x} _{1} \mathrm{y} _{2}+\mathrm{x} _{2} \mathrm{y} _{3}+\mathrm{x} _{3} \mathrm{y} _{1}\right)-\left(\mathrm{x} _{2} \mathrm{y} _{1}+\mathrm{x} _{3} \mathrm{y} _{2}+\mathrm{x} _{1} \mathrm{y} _{3}\right)\right] \end{aligned} $

1. If area of a triangle is given then use $\pm$ sign.

2. If three points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are collinear then area of triangle $\mathrm{ABC}$ is zero.

3. If $a _{1} x+b _{1} y+c _{1}=0 ; a _{2} x+b _{2} y+c _{2}=0$ and $a _{3} x+b _{3} y+c _{3}=0$ are the three sides of a triangle, then the area of the triangle is given by

$ \Delta=\frac{1}{2 \mathrm{C} _{1} \mathrm{C} _{2} \mathrm{C} _{3}}\left|\begin{array}{lll} \mathrm{a} _{1} & \mathrm{~b} _{1} & \mathrm{c} _{1} \\ \mathrm{a} _{2} & \mathrm{~b} _{2} & \mathrm{c} _{2} \\ \mathrm{a} _{3} & \mathrm{~b} _{3} & \mathrm{c} _{3} \end{array}\right|^{2} $

where $\mathrm{C} _{1}=\mathrm{a} _{2} \mathrm{~b} _{3}-\mathrm{a} _{3} \mathrm{~b} _{2}$

$ C _{2}=a _{3} b _{1}-a _{1} b _{3} $

$ C _{3}=a _{1} b _{1}-a _{2} b _{1} $

$\mathrm{C} _{1}, \mathrm{C} _{2}, \mathrm{C} _{3}$ are cofactors of $\mathrm{c} _{1}, \mathrm{c} _{2} \& \mathrm{c} _{3}$

Area of polygon

The area of polygon whose vertices are $\left(\mathrm{x} _{1}, \mathrm{y} _{1}\right),\left(\mathrm{x} _{2} \mathrm{y} _{2}\right) \ldots \ldots \ldots \ldots\left(\mathrm{x} _{\mathrm{n} 1} \mathrm{y} _{\mathrm{n}}\right)$

$ =\frac{1}{2}\left|\begin{array}{cc} \mathrm{x} _{1} & \mathrm{y} _{1} \\ \mathrm{x} _{2} & \mathrm{y} _{2} \\ \cdot & \\ \cdot & \\ \cdot & \\ \mathrm{x} _{\mathrm{n}} & \mathrm{y} _{\mathrm{n}} \\ \mathrm{x} _{1} & \mathrm{y} _{1} \end{array}\right|=\frac{1}{2}\left[\left(\mathrm{x} _{1} \mathrm{y} _{2}+\mathrm{x} _{2} \mathrm{y} _{3} \ldots \ldots \ldots \ldots \mathrm{x} _{\mathrm{n}} \mathrm{y} _{1}\right)-\left(\mathrm{y} _{1} \mathrm{x} _{2}+\mathrm{y} _{2} \mathrm{x} _{3} \ldots \ldots \ldots+\mathrm{y} _{\mathrm{n}} \mathrm{x} _{1}\right)\right] $

Examples

1. If the area of the rhombus by the lines $\mathrm{x} \pm \mathrm{my} \pm \mathrm{n}=0$ be 2sq. units then

(a) , $\mathrm{m}, \mathrm{n}$ are in GP

(b) , $\mathrm{n}, \mathrm{m}$ are in GP

(c) $\mathrm{m}=\mathrm{n}$

(d) $\mathrm{L}=\mathrm{m}$.

2. The area of the triangular region in the first quadrant bounded on the left by the $y$-axis; bounded above by the line $7 x+4 y=168$ and bounded below by the line $5 x+3 y=121$ is A, then the

value of $\frac{3 \mathrm{~A}}{10}$ is

3.

Show Answer

Solution:

a $\quad \rightarrow \mathrm{q}, \mathrm{r}, \mathrm{s}$

b $\quad \rightarrow \mathrm{p}$

c $\quad \rightarrow \mathrm{q}, \mathrm{s}$

d $ \rightarrow \mathrm{q}$

a. $\quad \mathrm{h} _{1}=\frac{10}{\sqrt{10}}=\sqrt{10}$

$\mathrm{h} _{2}=\frac{10}{\sqrt{10}}=\sqrt{10}$

Hence the given lines form a square of side $\sqrt{10}$. Area of a square is $(\sqrt{10})^{2}=10$ sq.units

b. slop of $\mathrm{AB}=\frac{5}{-1}=-5$

slop of $\mathrm{DC}=\frac{9}{-1}=-9$

slop of $\mathrm{AD}=\frac{-2}{3}=\frac{-2}{3}$

slop of $\mathrm{BC}=\frac{2}{3}$

Hence the figure is neither a parallelogram nor a trapezium

c. $\mathrm{h} _{1}=\frac{29}{\sqrt{58}}=\sqrt{\frac{29}{2}}$

$\mathrm{h} _{2}=\frac{29}{\sqrt{58}}=\sqrt{\frac{29}{2}}$

Hence given lines form a square of side $\sqrt{\frac{29}{2}}$ and area $\frac{29}{2}$ sq.units

d. $4 y-3 x-21=0$ and $4 y-3 x-7=0$ are parallel and $3 y-4 x+14=0$ and $3 y-4 x+7=0$ are parallel.

But $4 y-3 x-21=0$ and $3 y-4 x+14=0$ are not perpendicular

Hcnce the given line form a parallelogram.

4. The straight lines $7 \mathrm{x}-2 \mathrm{y}+10=0$ and $7 \mathrm{x}+2 \mathrm{y}-10=0$ form an isosceles triangle with the line $\mathrm{y}=2$.

Area of this triangle is equal to

(a) $\frac{15}{7}$ sq.units

(b) $\frac{10}{7}$ sq.units

(c) $\frac{18}{7}$ sq.units

(d) None

Show Answer

Solution:

$ \mathrm{B}\left(\frac{6}{7}, 2\right) \text { and } \mathrm{C}\left(\frac{-6}{7}, 2\right) $

$\mathrm{BC}=\frac{12}{7}$ and $\mathrm{AD}=3$

area of $\triangle \mathrm{ABC}=\frac{1}{2} \times \frac{12}{7} \times 3$

$ =\frac{18}{7} \text { sq.units. } $

Linked comprehension type

For problems 5 - 7

Let $\mathrm{ABCD}$ be a parallelogram whose equations for the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ are $\mathrm{x}+2 \mathrm{y}=3$ and $2 x+y=3$, respectively. If length of diagonal $A C=4$ units and area of parallelogram $A B C D=8$ sq.units. then

5. The length of other diagonal $\mathrm{BD}$ is

(a) 103

(b) 2

(c) $20 / 3$

(d) None

6. The length of side $\mathrm{AB}$ is equal to

(a) $\frac{2 \sqrt{58}}{3}$

(b) $\frac{4 \sqrt{58}}{9}$

(c) $\frac{3 \sqrt{58}}{9}$

(d) $\frac{5 \sqrt{58}}{9}$

7. The length of $\mathrm{BC}$ is equal to

(a) $\frac{2 \sqrt{10}}{3}$

(B) $\frac{4 \sqrt{10}}{3}$

(C) $\frac{8 \sqrt{10}}{3}$

(d) None

Show Answer

Solution:

5. $\tan \theta=\left|\frac{\frac{-1}{2}+2}{1+1}\right|=\left|\frac{3}{4}\right|$

$\sin \theta=\frac{3}{5}, \quad \cos \theta=\frac{4}{5}$

Area of $\Delta \mathrm{CPB}=\frac{1}{2} \mathrm{PC} \times \mathrm{PB} \sin \theta$

$\frac{1}{4} \times 8=\frac{1}{2} \times 2 \times \mathrm{PB} \times \frac{3}{5}$

$\mathrm{PB}=\frac{10}{3}$

$\mathrm{BD}=\frac{20}{3}$ units

6. $\cos (\pi-\theta)=\frac{\mathrm{AP}^{2}+\mathrm{PB}^{2}-\mathrm{AB}^{2}}{2 \mathrm{AP} \times \mathrm{PB}}$

$-\cos \theta=\frac{-4}{5}=\frac{4+\frac{200}{9}-\mathrm{AB}^{2}}{2 \times 2 \times \frac{10}{3}}$

$ \mathrm{AB}=\frac{2 \sqrt{58}}{3} $

7. In ${ } _{\Delta} \mathrm{CPB}, \cos \theta=\frac{P C^{2}+P B^{2}-B C^{2}}{2 \cdot P C \cdot P B}$

$\mathrm{BC}=\frac{2 \sqrt{10}}{3}$

8. Reasoning Type

Statement 1 : The area of the triangle formed by the points $\mathrm{A}(2000,2002), \mathrm{B}(2001,2004) \mathrm{C}(2002$, $2003)$ is same as the area formed by $\mathrm{A}^{1}(0,0), \mathrm{B}^{1}(1,2), \mathrm{C}^{1}(2,1)$

Statement 2 : The area of the triangle is constant with respect to translation of axes.

(a) both the statements are true but statement 2 is the correct explanation of statement 1

(b) both the statement are true but statement 2 is not the correct explanation of statement 1

(c) statement 1 is true and 2 is false

(d) statement 1 is false and 2 is true