চট্টগ্রাম বিশ্ববিদ্যালয় অনার্স ১ম বর্ষ (২০২২-২০২৩) ভর্তি বিজ্ঞপ্তি

strieght line;

 1. Distance formula:

d = √[(x2 – x1)2 + (y2 – y1)2]

2. Section Formula:

x = (mx2 + nx1)/(m + n)

y = (my2 + ny1)/(m + n)

3. Centroid:

G = [(x1 + x2 + x3)/3, (y1 + y2 + y3)/3]

4. Incentre:

I = {(ax1 + bx2 + cx3)/(a + b + c), (ay1 + by2 + cy3) / (a + b + c)}

5. Excentre:

I1 = {(-ax1 + bx2 + cx3)/(-a + b + c), (-ay1 + by2 + cy3)/(-a + b + c)}

6. Area of a Triangle:

Area of triangle ABC 

$\begin{array}{l}=\frac{1}{2}\left|\begin{array}{ccc}{�}_1& {�}_1& 1\\ {�}_2& {�}_2& 1\\ {�}_3& {�}_3& 1\end{array}\right|\end{array}$

7. Slope formula:

(i) Line joining two points (x1, y1) and (x2, y2), m = (y1  – y2) / (x1 – x2)

(ii) Slope of a line ax + by + c = 0 is -coefficient of x/coefficient of y = -a/b

8. Condition of collinearity of three points: 

$\begin{array}{l}\left|\begin{array}{ccc}{�}_1& {�}_1& 1\\ {�}_2& {�}_2& 1\\ {�}_3& {�}_3& 1\end{array}\right|=0\end{array}$

9. Equation of a straight line in various forms:

(i) Point Slope form: y – y1 = m(x – x1)

(ii) Slope intercept form: y = mx + c

(iii) Two point form: y – y1 = {(y2 – y1) / (x2 – x1)} × (x – x1)

(iv) Determinant form: 

$\begin{array}{l}\left|\begin{array}{ccc}�& �& 1\\ {�}_1& {�}_1& 1\\ {�}_2& {�}_2& 1\end{array}\right|=0\end{array}$

(v) Intercept form: (x/a) + (y/b) = 1

(vi) Perpendicular / Normal form: x cos α  + y sin α   = p

(vii) Parametric form: x = x1+ r cos θ   , y = y1 + r sin θ

(viii) Symmetric form: (x – x1)/cos θ  = (y – y1) / sin θ = r

(ix) General form: ax + by + c = 0

x intercept = -c/a

y intercept = -c/b

10. Angle between two straight lines in terms of their slopes:

$\begin{array}{l}\tan �=\left|\frac{{�}_1-{�}_2}{1+{�}_1{�}_2}\right|\end{array}$

11. Parallel lines:

Two lines ax + by + c = 0 and a’x + b’y + c’ = 0 are parallel if a/a’ = b/b’ ≠ c/c’.

Thus any line parallel to ax + by + c = 0 is of the type ax + by + k = 0, where k is a parameter.

If ax + by + c1 = 0 and ax + by + c2 = 0 are two parallel lines then the distance between these two parallel lines   

$\begin{array}{l}=\left|\frac{{�}_1-{�}_2}{\sqrt{{�}^2+{�}^2}}\right|\end{array}$

12. Perpendicular lines:

Two lines ax + by + c = 0 and a’x + b’y + c’ = 0 are perpendicular if aa’+bb’ = 0

13. Position of the points (x1, y1) and (x2, y2) relative to the line  ax + by + c = 0:

In general, two points (x1, y1) and (x2, y2) will lie on the same side or opposite side of ax + by + c = 0 according to as ax1 + by1 + c and ax2 + by2 + c are of the same or opposite sign, respectively.

14. Length of the perpendicular from a point on a line :

The length of the perpendicular from a point (x1, y1) to a line ax + by + c = 0 is

$\begin{array}{l}\left|\frac{�{�}_1+�{�}_1+�}{\sqrt{{�}^2+{�}^2}}\right|\end{array}$

15. Reflection of a point about a line:

(i) Foot of the perpendicular from a point on the line is (x – x1)/a = (y – y1)/b = -(ax1 + by1 + c)/(a2 + b2)

(ii)Image of (x1, y1) in the line ax + by + c = 0 is (x – x1)/a = (y – y1)/b = -2 (ax1 + by1 + c)/(a2 + b2)

16. Bisectors of the angles between two lines: 

$\begin{array}{l}\frac{��+��+�}{\sqrt{{�}^2+{�}^{{}^2}}}=\pm \frac{{�}^{\prime }�+{�}^{\prime }�+{�}^{\prime }}{\sqrt{{�}^{\prime 2}+{�}^{\prime {}^2}}}\end{array}$

17. Methods to discriminate between the acute angle bisector and the obtuse angle bisector:

If aa’ + bb’<0, cc’ > 0, then the equation of the bisector of acute angle is:

$\begin{array}{l}\frac{��+��+�}{\sqrt{{�}^2+{�}^{{}^2}}}=+\frac{{�}^{\prime }�+{�}^{\prime }�+{�}^{\prime }}{\sqrt{{�}^{\prime 2}+{�}^{\prime {}^2}}}\end{array}$

If aa’ + bb’> 0, cc’ > 0, then the equation of the bisector of obtuse angle is:

$\begin{array}{l}\frac{��+��+�}{\sqrt{{�}^2+{�}^{{}^2}}}=\frac{{�}^{\prime }�+{�}^{\prime }�+{�}^{\prime }}{\sqrt{{�}^{\prime 2}+{�}^{\prime {}^2}}}\end{array}$

18. Discriminate between the bisector of the angle containing a point:

To discriminate between the bisector of the angle containing the origin and that of the angle not containing the origin. Rewrite the equations, ax + by + c = 0 and a’x + b’y + c’ = 0 such that the constant terms c and c’ are positive.

Then;

$\begin{array}{l}\frac{��+��+�}{\sqrt{{�}^2+{�}^{{}^2}}}=–\frac{{�}^{\prime }�+{�}^{\prime }�+{�}^{\prime }}{\sqrt{{�}^{\prime 2}+{�}^{\prime {}^2}}}\end{array}$

gives the equation of the bisector of the angle not containing the origin.

In general, equation of the bisector which contains the point is 

$\begin{array}{l}\frac{��+��+�}{\sqrt{{�}^2+{�}^{{}^2}}}=+\frac{{�}^{\prime }�+{�}^{\prime }�+{�}^{\prime }}{\sqrt{{�}^{\prime 2}+{�}^{\prime {}^2}}}\end{array}$

Or 

$\begin{array}{l}\frac{��+��+�}{\sqrt{{�}^2+{�}^{{}^2}}}=–\frac{{�}^{\prime }�+{�}^{\prime }�+{�}^{\prime }}{\sqrt{{�}^{\prime 2}+{�}^{\prime {}^2}}}\end{array}$

 according to a + b + c and a’ + b’ + c’ having the same sign or otherwise.

19. Condition of congruency: of three straight lines aix + biy + ci = 0, i = 1, 2, 3 is

$\begin{array}{l}\left|\begin{array}{ccc}{�}_1& {�}_1& {�}_1^{}\\ {�}_2& {�}_2& {�}_2\\ {�}_3& {�}_3& {�}_3\end{array}\right|=0\end{array}$

20. Family of straight lines:

The equation of a family of straight lines passing through the point of intersection of the lines,

L1 = a1x + b1y + c1 = 0 and L2 = a2x + b2y + c2 = 0 is given by L1 + kL2 = 0

21. A pair of straight lines through origin: ax2 + 2hxy + by2 = 0

If θ is the acute angle between the pair of straight lines, then 

$\begin{array}{l}\tan �=\left|\frac{2\sqrt{{ℎ}^2-��}}{�+�}\right|\end{array}$

22. General equation of second degree representing a pair of straight lines:

ax2 + 2hxy + by2 + 2gx + 2fy + c = 0 represents a pair of straight lines if abc +2fgh – af2 – bg2 – ch2 = 0

i.e. if

$\begin{array}{l}\left|\begin{array}{ccc}�& ℎ& �\\ ℎ& �& �\\ �& �& �\end{array}\right|=0\end{array}$

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