COORDINATE GEOMETRY - 4 (Straight Line)

Topics Covered

1. General Equation of straight line

2. Reducing general equations to

i. Slope intercept form

ii. Intercept form

iii. Normal form

1. General Equation of straight line

First degree equation of the form $a x+b y+c=0$ where $a, b, c \in R$ and $b$ can be zerobut not both at the same time.

2. Reducing general equation to

i. Slope intercept form

Given equation is $a x+b y+c=0$

Rewrite the equation by $=-\mathrm{ax}-\mathrm{c}$

divide by $\mathrm{b}$ we get $-\frac{\mathrm{a}}{\mathrm{b}} \mathrm{x}-\frac{\mathrm{c}}{\mathrm{b}}$

It look like $y=m x+c$

slope $=\frac{-a}{b}=-\frac{\text { Coefficient of } x}{\text { Coefficient of } y}=m$

$\mathrm{y}$ - intereept $=\frac{-\mathrm{c}}{\mathrm{b}}=-\frac{\text { Constant }}{\text { Coefficient of } \mathrm{y}}$

ii. Reducing to intercept form

Given equation is $a x+b y+c=0$

Rewrite the equation $a x+b y=-c$

$ \begin{aligned} & \frac{x}{-c / a}+\frac{y}{-c / b}=1 \\ & \frac{x}{a}+\frac{y}{b}=1 \end{aligned} $

So intercepts are -c/a & -c/b on x-axis and y-axis respectively.

iii. Reduce to normal form

Given equation be

Re write the equation

$ \begin{aligned} & a x+b y+c=0 \\ & a x+b y=-c \\ & -a x-b y=c \end{aligned} $

Keeping constant term postive

Divide by $\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}$ we get $-\frac{a}{\sqrt{a^{2}+b^{2}}} x-\frac{b}{\sqrt{a^{2}+b^{2}}} y=\frac{c}{\sqrt{a^{2}+b^{2}}}$

It look like $\mathrm{x} \cos \alpha+\mathrm{y} \sin \alpha=\mathrm{p}$

Where $\cos \alpha=\frac{-\ {a}}{\sqrt{\ {a}^{2}+\ {b}^{2}}}, \sin \alpha=\frac{-\ {b}}{\sqrt{\ {a}^{2}+\ {b}^{2}}}$ (or) $\tan \alpha=\frac{\ {b}}{\ {a}}$

distance of a line from the origin is always positive

$\therefore \frac{\mathrm{c}}{\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}}=\mathrm{p}$ is positive.

Practice questions

1. A triangle $\mathrm {ABC}$ with vertices $\mathrm{A}(-1,0), \mathrm{B}\left(-2, \frac{3}{4}\right) \& \mathrm{C}\left(-3,-\frac{7}{6}\right)$ has its orthocenter $\mathrm{H}$. Then the orthocentre of triangle $\mathrm{BCH}$ will be

(a) $(-1,0)$

(b) $(-3,-2)$

(c) $(1,3)$

(d) $(-1,2)$

2. The points $\mathrm{A}(0,0), \mathrm{B}(\cos \alpha, \sin \alpha)$ and $\mathrm{C}(\cos \beta, \sin \beta)$ are the vertices of a right angled triangle if

(a) $ \sin \left(\frac{\alpha+\beta}{2}\right)=\frac{1}{\sqrt{2}}$

(b) $ \cos \left(\frac{\alpha+\beta}{2}\right)=\frac{1}{\sqrt{2}}$

(c) $ \cos \left(\frac{\alpha-\beta}{2}\right)=\frac{1}{\sqrt{2}}$

(d) $ \sin \left(\frac{\alpha+\beta}{2}\right)=-\frac{1}{\sqrt{2}}$

3. Set of values of $\alpha$ for which the point $\left(\alpha, \alpha^{2}-2\right)$ lies inside the triangle formed by the lines $\mathrm{x}+\mathrm{y}=$ $1, \mathrm{y}=\mathrm{x}+1$ and $\mathrm{y}=-1$ is

(a) $\left(\frac{1-\sqrt{13}}{2},-1\right) \mathrm{U}\left(1, \frac{-1+\sqrt{13}}{2}\right)$

(b) $(1, \sqrt{13})$

(c) $(-\sqrt{13},-1)$

(d) None

4. Area of the parallelogram formed by the lines $y=m x, y=m x+1, y=n x+1$ and $y=n x$ equals

(a) $\frac{|\mathrm{m}+\mathrm{n}|}{(\mathrm{m}-\mathrm{n})^{2}}$

(b) $\frac{2}{|m+n|}$

(c) $\frac{1}{|m+n|}$

(d) $\frac{1}{|m-n|}$

5. $\mathrm{A}$ and $\mathrm{B}$ are fixed points. The vertex $\mathrm{C}$ of $\triangle \mathrm{ABC}$ moves such that $\cot \mathrm{A}+\cot \mathrm{B}=\operatorname{constant}$. The locus of $C$ is a straight line

(a) perpendicular to $\mathrm{AB}$

(b) parallel to $\mathrm{AB}$

(c) inclined at an angle (A-B) to AB

(d) None of these.

6. The area of the figure formed by a|x $|+b| y \mid+c=0$ is

(a) $\frac{c^{2}}{|a b|}$

(b) $\frac{2 \mathrm{c}^{2}}{|\mathrm{ab}|}$

(c) $\frac{c^{2}}{2|a b|}$

(d) None of these

7. The orthocentre, circumcentre, centroid and incentre of the triangle formed by the line $\mathrm{x}+\mathrm{y}=\mathrm{a}$ with the coordinate axes lie on

(a) $\mathrm{x}^{2}+\mathrm{y}^{2}=1$

(b) $y=x$

(c) $y=2 x$

(d) $y=3 x$

8. Two points $(a, 3)$ and $(5, b)$ are the opposite vertices of a rectangle. If the other two vertices lie on the line $y=2 x+c$ which passes through the point $(a, b)$ then the value of $c$ is

(a) $-7$

(b) $-4$

(c) $0$

(d) $7$