HSC-Geometry Equation -HSC & Honours (Short Technique & Formulas) by Tanbircox

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x (tan)

m

AB x 000 90;1800

AB m = tan .

PQ x 0180

PQ , 0180tanm = tan . 00 18090



PQ x 11 y,xA 22 y,xB

PQ BARtantanm 21

21

12

12

xx

yy

xx

yy

AR

BR

= /

: 332211 y,xC,y,xB,y,xA :

321

31

31

21

21 x,xx;xx

yy

xx

yy

x by

y - ax

: b x b b x b

b=0 x

x 0y



a y a a y a

a = 0 y

y

PQ y c x

, A (x,y) PQ y OC= C.

cmxy

0c PQ

mxy

11 y,x 11 xxmyy

11 y,x 22 y,x

21

1

21

1

yy

yy

xx

xx

(i) (0,0) 11 y,x : xxy

yy

y

x

x

1

1

11

(ii) = 12

12

xx

yy



PQ y,xA x P y- Q

OP = a OQ = b 1b

y

a

x

(0,0)

Psinycosx

11 y,x

sin

yy

cos

xx 11

0cbyax 0cybxa 111

111 cc,bb,aa

111 c

c

b

b

a

a



1. A (x1,y1) B (x2,y2) (gradient) ,

2. ax+by+c=0 , m = -(a/b)

3. A (x1, y1), B (x2, y2) C (x3, y3) AB AC

,

4. x , y = 0

5. y , x = 0

6. x , y = b

7. y , x = a

8. y c x , y = mx+c , m = = tan c = 0 , y = mx

9.(x1,y1) m y-y1 = m(x-x1)



10.(x1, y1) (x2,y2) y , =

11. (0,0) (x1,y1) , (x/x1) = (y/y1)

12.x a y b

, x/a + y/b = 1

x (a,0) y (0,b)

13. x

p , x cos + ysin = p

14.

15.a1x+b1y+c1 = 0 a2x+b2y+c2 = 0 ,

a1x+b1y+c1+k(a2x+b2y+c2) = 0

k-

16. (x1, y1) (x2,y2) ax+by+c = 0 a1x+b1y+c a2x+b2y+c

17. (x1, y1) (x2,y2) ax+by+c = 0 a1x+b1y+c a2x+b2y+c



18. m1 m2 m1m2 = -1 m1= m2

19. a1x+b1y+c1 = 0 a2x+b2y+c2 = 0 a1a2+b1b2 = 0

(a1/b1) = (a2/b2)

20. m1 m2 / , tan =

tan

21. a1x+b1y+c1 = 0 a2x+b2y+c2 = 0 ,

tan =

tan

22.ax+by+c1 = 0 , ax+by+c2 = 0 ,

23.(x1,y1) ax+by+c = 0 , a(x-x1)+b(y-y1) = 0

24.ax+by+c1 = 0 , bx-ay+c2 = 0 ,x y

,

25. (x1,y1) ax+by+c = 0 , b(x-x1) a(y-y1) = 0

26. a1x+b1y+c1 = 0; a2x+b2y+c2 = 0 a3x+b3y+c3 = 0 , = 0



27. =

, D = C1, C2 , C3 c1, c2, c3

28.ax+by+c = 0 (x1,y1) , d =

29. ax+by+c1 = 0 ax+by+c2 = 0 , d =

30. a1x+b1y+c1 = 0 a2x+b2y+c2 = 0 ,

1. a1a2+b1b2 > 0 + -

2. a1a2+b1b2 < 0 +



. (0,0) r

x2+y2 = ry2

. (h,k) r (x-h)2+(y-k)2 = r2

h=0 y , x2+(y-k)2=k2 k=0 x , (x-h)2+y2=h2

. , x2+y2+2gx+2fy+c=0

, (-g,-f) = (g2+f2-c) g = 0 y f = 0 x c = 0



. x x = 2(g2-c) x g2=c

y y = 2(f2-c) y f2=c

. x

, (x-h)2+(y-k)2 = k2

. y

, (x-h)2+(y-k)2 = h2

, (x1,y1) (x2,y2)

, (x-x1)(x-x2)+(y-y1)(y-y2) = 0

. x2+y2+2gx+2fy+c=0 , x2+y2+2gx+2fy+c1=0

. x2+y2+2gx+2fy+c=0 ax+by+c1 ,

x2+y2+2gx+2fy+c+k(ax+by+c1)=0

. , =



. , =

.

.

. x2+y2+2gx+2fy+c=0 x2+y2+2g1x+2f1y+c1=0 ,

x2+y2+2gx+2fy+c+k(x2+y2+2g1x+2f1y+c1)=0

.



. y=mx+c x2+y2 = r2 , c = r(1+m2)

. x2+y2=r2 (x1,y1) , xx1+yy1=r2

. x2+y2+2gx+2fy+c = 0 (x1,y1) ,

xx1+yy1+g(x+x1)+f(y+y2)+c = 0

. (x1,y1) x2+y2 = r2

, (x2+y2-r2)(x12+y12-r2)=(xx1+yy1-r2)2

. (x1,y1) x2+y2+2gx+2fy+c=0 ,

(x2+y2+2gx+2fy+c)(x12+y12+2gx1+2fy1+c) = {xx1+yy1+g(x+x1)+f(y+y1)+c}

. (x1, y1) x2+y2=a2 , = (x2+y2-r2) x2+y2+2gx+2fy+c=0 , = (x12+y12+2gx1+2fy1+c)

. x2+y2 = r2 (x1,y1) , x1y-y1x=0

. x2+y2+2gx+2fy+c=0 (x1,y1) ,

(x1+g)y-(y1+f)x+fx1-gy1=0

. x2+y2+2g1x+2f1y+c1 = 0 x2+y2+2g2x+2f2y+c2 = 0 ,

(x2+y2+2g1x+2f1y+c1) (x2+y2+2g2x+2f2y+c2)=0



,

(focus) (directrix) (eccentricity) e e

e = 0 (circle) 0 < e < 1 (ellipse) e = 1 (parabola) e > 1 (hyperbola)

(Parabola) :



r r

(Unit vector) : (Magnitude)

A ,

(Rectangular unit vectors) : x, y z , , k



O P ,

;

;

A = Ax+ Ay + Azk ; B = Bx + By + Bzk A + B = (Ax+Bx) + (Ay+By) + (Az+Bz) k

C = Cx + Cy+ Czk [ C = ]

P Q ,

R = P +Q

R ,P ,

A B

, ,



. B = ABcos [A . B = B . A ] , A = Ax+ Ay + Azk ;

B = Bx + By + Bzk , A . = AxBx + AyBy + AzBz A B = 90 A . B = AB cos 90 = 0 [cos90 = 0] ,

A B

,

C = A B = ABsin [A B B A ] C , A = Ax+ Ay + Azk ; B = Bx + By + Bzk ,

A B , = 0 A B = AB sin0 = 0 [sin0 = 0] ,

A B ,

,

A B =

B A =



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HSC-Geometry Equation -HSC & Honours (Short Technique & Formulas) by Tanbircox

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