Distance Formula :

PYQ-2023-Straight-Line-Q2, PYQ-2023-Straight-Line-Q4, PYQ-2023-Straight-Line-Q7, PYQ-2023-Circle-Q1, PYQ-2023-Ellipse-Q1, PYQ-2023-Ellipse-Q5

Given two points $ \ P_1(x_1, y_1) \ $ and $ \ P_2(x_2, y_2) \ $ the distance ( d ) between these points is given by:

$$ d=\sqrt{\left(x_{1}-x_{2}\right)^{2}+\left(y_{1}-y_{2}\right)^{2}} $$

Section Formula :

PYQ-2023-Straight-Line-Q3, PYQ-2023-circle-Q2

(i) Internal section formula :

$$P(x, y)=\left(\frac{m x_2+n x_1}{m+n}, \frac{m y_2+n y_1}{m+n}\right)$$

(ii) External section formula :

$$P(x, y)=\left(\frac{m x_2-n x_1}{m-n}, \frac{m y_2-n y_1}{m-n}\right)$$

Centroid, Incentre and Excentre:

PYQ-2023-Straight-Line-Q2

Centroid (G) $$(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} )$$

Incentre $$(\frac{ax_1 + bx_2 + cx_3}{a+b+c}, \frac{ay_1 + by_2 + cy_3}{a+b+c} )$$

Excentre (I_{1}) $$ (\frac{-ax_1 + bx_2 + cx_3}{-a+b+c}, \frac{-ay_1 + by_2 + cy_3}{-a+b+c} )$$

Area of a Triangle:

PYQ-2023-Straight-Line-Q1, PYQ-2023-Straight-Line-Q7, PYQ-2023-Circle-Q7, PYQ-2023-Parabola-Q6, PYQ-2023-Ellipse-Q3

The area of a triangle with vertices $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$, $Q\left(x_2, y_2\right)$, and $R\left(x_3, y_3\right)$

$$ \triangle \text{ABC} = \frac{1}{2}\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right|$$

Slope Formula:

PYQ-2023-Straight-Line-Q5

Line joining two points $(x_1 , y_1)$ & $(x_2 , y_2)$

$$m=\frac{y_1-y_2}{x_1-x_2}$$

Equation Of A Straight Line In Various Forms:

PYQ-2023-Straight-Line-Q3, PYQ-2023-Circle-Q3, PYQ-2023-Circle-Q8, PYQ-2023-3D-Q2, PYQ-2023-3D-Q4, PYQ-2023-3D-Q8, PYQ-2023-3D-Q12

(i) Point Slope form: $$y-y_1=m\left(x-x_1\right)$$

(ii) Slope intercept form: $$y=m x+c$$

(iii) Two point form: $$y - y_1 = \left(\frac{y_2 - y_1}{x_2 - x_1}\right)(x - x_1)$$

(iv) Intercept form: $$\frac{x}{a} + \frac{y}{b} = 1$$

(v) Perpendicular / Normal form: $$x \cos \alpha+y \sin \alpha=p$$ PYQ-2023-Straight-Line-Q8

(vi) Parametric form: $$x=x_1+r \cos \theta, y=y_1+r \sin \theta$$

(vii) Symmetric form: $$\frac{x - x_1}{\cos \theta} = \frac{y - y_1}{\sin \theta} = r $$

(viii) General form: $$a x+b y+c=0$$

Angle Between Two Straight Lines :

$$\tan \theta=\left|\frac{m_{1}-m_{2}}{1+m_{1} m_{2}}\right| $$

Condition for Parallel or Perpendicular of two lines: $a x+b y+c=0$ and $a^{\prime} x+b^{\prime} y+c^{\prime}=0$

PYQ-2023-Straight-Line-Q9, PYQ-2023-Circle-Q2, PYQ-2023-Parabola-Q1, PYQ-2023-Ellipse-Q3

$ \quad $ (i) parallel if $$\frac{\mathrm{a}}{\mathrm{a}^{\prime}}=\frac{\mathrm{b}}{\mathrm{b}^{\prime}} \neq \frac{\mathrm{c}}{\mathrm{c}^{\prime}}$$

$ \quad $ (ii) Perpendicular If $$\mathrm{aa}^{\prime}+\mathrm{bb}^{\prime}=\mathbf{0}$$

$ \quad $ (iii) Distance between two parallel lines $$d=\left|\frac{c_{1}-c_{2}}{\sqrt{a^{2}+b^{2}}}\right|$$

Condition of Collinearity of Three Points:

$$\left|\begin{array}{lll}x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1\end{array}\right|=0$$

A Point and Line:

PYQ-2023-Straight-Line-Q6, PYQ-2023-Circle-Q4 , PYQ-2023-Circle-Q8, PYQ-2023-Parabola-Q3, PYQ-2023-Parabola-Q5, PYQ-2023-Ellipse-Q4, PYQ-2023-3D-Q10, PYQ-2023-3D-Q27

(i) Distance between point and line $$d=\left|\frac{a x_{1}+b y_{1}+c}{\sqrt{a^{2}+b^{2}}}\right|$$

(ii) Reflection of a point about a line: $$\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{-2(a x_{1}+b y_{1}+c)}{a^{2}+b^{2}}$$

(iii) Foot of the perpendicular from a point on the line

$$\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=-\frac{a x_{1}+b y_{1}+c}{a^{2}+b^{2}}$$

Bisectors Of The Angles Between Two Lines:

PYQ-2023-Circle-Q6

$$\frac{a x+b y+c}{\sqrt{a^{2}+b^{2}}}= \pm \frac{a^{\prime} x+b^{\prime} y+c^{\prime}}{\sqrt{a^{\prime 2}+b^{\prime 2}}}$$

Condition Of Concurrency Of Three Straight Lines $a_{i} x+b_{i} y+c_{i}=0, i=1,2,3$

$$ \left|\begin{array}{lll}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|=0$$

A Pair Of Straight Lines Through Origin:

$$ ax^2 + 2hxy + by^2 = 0 $$

If $\theta$ is the acute angle between the pairs of the straight line ,then

$$ \tan \theta = \left|\frac{2 \sqrt{h^2 - ab}}{a + b}\right| $$

Equation Of Lines Parallel And Perpendicular To A Given Line :

(i) Equation of line parallel to line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$

$$a x+b y+\lambda=0 $$

(ii) Equation of line perpendicular to line $\mathrm{ax}+\mathrm{by}+\mathrm{c}=0$

$$ b x-a y+k=0 $$

Here $\lambda, \mathrm{k}$, are parameters

Family Of Lines :

If equation of two lines be $P \equiv a_1 x+b_1 y+c_1=0$ and $Q \equiv a_2 x+b_2 y+c_2=0$, then the equation of the lines passing through the point of intersection of these lines is :

$$\mathrm{P}+\lambda \mathrm{Q}=0$$ or $$\mathrm{a}_1 \mathrm{x}+\mathrm{b}_1 \mathrm{y}+ c_1+\lambda\left(a_2 x+b_2 y+c_2\right)=0$$

General Equation And Homogeneous Equation Of Second Degree :

PYQ-2023-Straight-Line-Q10

(i) A general equation of second degree ${a x}^2+{2 h x y}+{b y}^2+{2 g x}+{2 f y}+{c}={0}$ represent a pair of straight lines if $$\Delta=a b c+2 f g h-a f^2-b^2-c h^2=0$$ or

$$\left|\begin{array}{lll}a & h & g \\ h & b & f \\ g & f & c \end{array}\right|=0 $$

(ii) If $\theta$ be the angle between the lines, then $$\tan \theta= \pm \frac{2 \sqrt{\mathrm{h}^2-\mathrm{ab}}}{\mathrm{a}+\mathrm{b}}$$ these lines are

• Parallel if $$\Delta=0, \mathrm{~h}^2=\mathrm{ab}$$ or $$\mathrm{h}^2=\mathrm{ab} , \mathrm{bg}^2=\mathrm{af}^2$$

• Perpendicular if $$a+b=0$$ i.e. $$ \text{ coefficient of}\quad x^2+ \text{coefficient of}\quad y^2=0 $$

(iii) Homogeneous equation of 2nd degree $ ax^2 + 2hxy + by^2 = 0 $ always represents a pair of straight lines whose equations are

$$ y = \left(\frac{-h \pm \sqrt{h^2 - ab}}{b}\right) x $$

or

$$y = m_1x , y = m_2x $$

and $$\mathrm{m}_1+\mathrm{m}_2=-\frac{2 \mathrm{~h}}{\mathrm{~b}} ; \mathrm{m}_1 \mathrm{~m}_2=\frac{\mathrm{a}}{\mathrm{b}}$$

These straight lines passes through the origin and for finding the angle between these lines same formula as given for general equation is used

The condition that these lines are:

• At right angles to each other is $$\mathrm{a}+\mathrm{b}=0$$ i.e. co-efficient of $x^2+$ co-efficient of $y^2=0$.

• Coincident $$h^2=a b$$

• Equally inclined to the axis of $x$ is $$h=0$$ i.e. coeff. of $$\mathrm{xy}=0$$

(iv) The combined equation of angle bisectors between the lines represented by homogeneous equation of $2^{\text {nd }}$ degree is given by $$\frac{x^2-y^2}{a-b}=\frac{x y}{h}, a \neq b, h \neq 0$$