1. Point in Cartesian Plane:

$x =$ Abscissa of any point; $y =$ ordinate of any point. Any point in cartesian plane is represented by $P (x, y)$ .

2. Distance between Two Points and Section Formula:

(i) Distance between point $P (x, y)$ and origin $O (0,0)$ is $O P = \sqrt{x^{2} + y^{2}}$ .

(ii) Distance between two points $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2})$ is $P Q = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ _.

(iii) Section formula for internal division if point $P$ divides the line segment joining by point $(x_{1}, y_{1})$ & $(x_{2}, y_{2})$ internally in ratio $m : n$ is $P (x, y) = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})$ _.

(iv) Section formula for External division of above cases is $Q (x, y) = (\frac{m x_{2} - n x_{1}}{m - n}, \frac{m y_{2} - n y_{1}}{m - n})$ _.

(v) Midpoint formula $P (x, y) = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$ _.

(vi) Harmonic conjugate of two points will be points which divide the line segments internally and externally in the same ratio.

3. Special Points in a Triangle:

(i) Orthocentre is the intersection point of the altitudes of a triangle.

(ii) Circumcenter is the intersection point of perpendicular bisectors of the sides of a triangle. Circumcenter of a triangle is equidistant from all the vertices.

(iii) Centroid is the intersection point of the medians of a triangle. Centroid of triangle with vertices $(x_{1}, y_{1}), (x_{2}, y_{2})$ and $(x_{3}, y_{3}) i s G (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})$ .

(iv) Incentre is the intersection point of the internal angle bisectors of a triangle. Incentre of triangle with vertices $(x_{1}, y_{1}) (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ is $(\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c})$ .

4. Equation of Straight Lines in Different Form

(i) Slope – intercept form: $y = m x + c$ , where $m$ is the slope of line and $c$ is $y$ -intercept of the line.

(ii) Point slope form: $y - y_{1} = m (x - x_{1})$ .

(iii) Two points form: $y - y_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} (x - x_{1})$ or $\frac{y - y_{1}}{y_{2} - y_{1}} = \frac{x - x_{1}}{x_{2} - x_{1}}$ .

(iv) Intercept form: $\frac{x}{a} + \frac{y}{b} = 1$ , where $a$ & $b$ are intercepts on coordinate axes.

(v) Normal form: $x c o s α + y s i n α = p$ where $p$ is the length of perpendicular drawn on the line from origin and $α$ is the angle made by perpendicular with positive direction of $x$ -axis. The line $a x + b y = c$ can be converted to normal form by dividing $\sqrt{a^{2} + b^{2}}$ _.

(vi) The parametric equation of a straight line is $\frac{x - x_{1}}{\cos θ} = \frac{y - y_{1}}{\sin θ} = \pm r$ , where $θ$ is the angle of inclination and $(x_{1}, y_{1})$ is the given point on the line.

(vii) General equation of straight line (linear equation) in $x$ and $y$ is $a x + b y + c = 0, a, b, c$ are constants. If $m$ is slope of this line, $m = \frac{- a}{b}$ _.

(viii) Equation of $x$ – axis is $y = 0$ _.

(ix) Equation of a line Parallel to $x$ – axis is $y = k,$ where $k$ is a constant.

(x) Equation of $y$ – axis is $x = 0$ _.

(xi) Equation of a line Parallel to $y$ – axis is $x = k$ , where $k$ is a constant.

(xii) Any point on $x$ – axis is $(a, 0)$ and on $y$ – axis is $(0, b)$ .

5. Interaction between Two Lines

(i) Acute angle between two lines with slopes $m_{1} & m_{2}$ is given by $\tan θ = |\frac{m_{1} - m_{2}}{1 + m_{1} m_{2}}|$ _.

(ii) The lines $a_{1} x + b_{1} y + c_{1} = 0 a n d a_{2} x + b_{2} y + c_{2} = 0$ are said to be

(a) Parallel if $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}}$

(b) Perpendicular if $a_{1} a_{2} + b_{1} b_{2} = 0$

(iii) Any line parallel to line $a x + b y + c = 0$ is of the form $a x + b y + k = 0$ where $k$ is a constant.

(iv) Any line perpendicular to line $a x + b y + c = 0$ is of form $b x - a y + k = 0$ where $k$ is a constant.

(v) If lines $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ intersects each other, then the point of intersection will be the solution of these two lines.

6. Pair of Straight Line:

Equation of pair of straight line will be in the form of $a x^{2} + b y^{2} + 2 g x + 2 f y + 2 h x y + c = 0$ , where $a b c + 2 f g h - a f^{2} - b g^{2} - c h^{2} = 0$ .

Homogeneous equation of pair of straight lines will be $a x^{2} + b y^{2} + 2 h x y = 0$ .

If slope of the lines are $m_{1}$ and $m_{2}$ then $m_{1} + m_{2} = - \frac{2 h}{b}, m_{1} . m_{2} = \frac{a}{b}$ .

7. Concurrency of Three Lines:

Three lines are said to be concurrent if they are meeting at the same point.

For this find the intersection point of any two lines, this should be satisfied by the equation of third line.

Three lines are concurrent if the value of their determinant is zero.

8. Family of Lines:

(i) The lines $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ are represented by $L_{1} = 0$ & $L_{2} = 0$ , then family of lines passing through intersection point of these two lines is $L_{1} + k L_{2} = 0$ , where $k$ is a parameter.

9. Locus Problems:

Locus is the path traced by a moving point under certain Geometrical conditions. Take any one point on the path of the given points and apply the given condition then change in general form that will be locus of points.

10. Shift of Origin:

(i) If origin $(0,0)$ shifted at point $(h, k)$ then coordinate of point $(x, y)$ will be $(x - h, y - k)$ and accordingly the equation of line will also be changed.

(ii) Similarly, transformation of axes will change coordinate of point and equation of line will be changed.

If origin remains same and coordinate axis rotates by the angle of A in anti-clockwise then new form $(x^{'}, y^{'})$ of coordinate of $(x, y)$ will be changed accordingly.

$x^{'} = x \cos A - y \sin A, y^{'} = - x \sin A + y \cos A$ .

11. Equation of Bisector of Angles between Two Lines:

The lines $a x + b y + c = 0$ and $a^{'} x + b^{'} y + c^{'} = 0$ then equation of their, angle bisectors will be $\frac{a x + b y + c}{\sqrt{a^{2} + b^{2}}} = \pm (\frac{a^{'} x + b^{'} y + c^{'}}{\sqrt{{a^{'}}^{2} + {b^{'}}^{2}}})$ .

If $a a^{'} + b b^{'}$ is $> 0$ then take $+$ for acute angle bisector and $-$ for obtuse angle bisector and if $a a^{'} + b b^{'}$ is $< 0$ then take $+$ for obtuse angle bisector and $-$ for acute angle bisector, where $c c^{'} > 0$ .

12. Straight Line and Point:

(i) Inclination $θ$ is the angle made by a line with $+$ ve direction of $x$ -axis in anticlockwise direction.

$(i i)$ Slope or gradient of a line is tangent of angle made the line with $+$ ve direction of $x$ -axis i.e. $m = \tan θ$

(iii) Slope of $x$ -axis is $0$ .

(iv) Slope of line parallel to $x$ -axis is $0$ .

(v) Slope of the $y$ -axis is not defined but for calculation purpose we take it as infinity.

(vi) Slope of line parallel to $y$ -axis is not defined.

(vii) If the lines are parallel, then $m_{1} = m_{2}$ _.

(viii) If the lines are perpendicular, then $m_{1} m_{2} = - 1$ _.

(ix) Slope of line joining two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ . If a straight line cuts $x$ -axis at $A$ and $y -$ axis at $B$ then $O A$ is called $x$ - intercept and $O B$ is called $y -$ intercept. Intercept can be $-$ ve or $+$ ve.

(x) Area of triangle with vertices $(x_{1}, y_{1}) (x_{2}, y_{2})$ and $(x_{3}, y_{3})$ is $A = \frac{1}{2} |Σ x_{1} (y_{2} - y_{3})|$ or $A = \frac{1}{2} |[x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2})]|$ _.

(xi) If one of the vertex is origin, then $Δ = \frac{1}{2} |x_{1} y_{2} - x_{2} y_{1}|$ _.

(xii) Three points are said to be collinear if $Δ = 0$ _.

(xiii) Length of perpendicular from $(x_{1}, y_{1})$ to line $a x + b y + c = 0$ is $\frac{|a x + b y + c|}{\sqrt{a^{2} + b^{2}}}$ .

(xiv) Distance between two parallel lines will be $\frac{|d_{1} - d_{2}|}{{(a^{2} + b^{2})}^{\frac{1}{2}}}$ .

(xv) Image of the point $(x_{1}, y_{1})$ about the line $a x + b y + c = 0$ will be $\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b}$ = $\frac{- 2 (a x_{1} + b y_{1} + c)}{a^{2} + b^{2}}$ .

(xvi) The points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ lie on the same side or opposite side of the line $a x + b y + c = 0$ , if $\frac{a x_{1} + b y_{1} + c}{a x_{2} + b y_{2} + c} > 0$ or $\frac{a x_{1} + b y_{1} + c}{a x_{2} + b y_{2} + c} < 0$ respectively.

y=mx be a variable line (m is variable) intersect the lines 2 x+y-2=0 and x-2 y+2=0 in P and Q. The locus of midpoint of segment of PQ. is

Important Questions on Straight Lines

Learn and Prepare for any exam you want !

$y = m x$ be a variable line ( $m$ is variable) intersect the lines $2 x + y - 2 = 0$ and $x - 2 y + 2 = 0$ in $P$ and $Q$ . The locus of midpoint of segment of $P Q$ . is