mathematics ge review session 5

8/15/2019 Mathematics GE REVIEW Session 5

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MATHEMATICSWith Casio fx 991-ES plus Calculator Techniques

SESSION 5

Engr. Broddett B. Abatayo, GE, REA


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CIRCLE


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CIRCLE

Ax² + By² + Cx + Dy + E !

$eneral Equation%

&A²

C²+D²-&AE%

'A

-C%

'A

-D

CIRCLETT’s formula

h

k r


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EXAMPLE 35% $i(en x² + y² - 1!x ) 1!y + ',in the center

a .*/-&0 .*/*0 c .2/-*0 .&/-*

CIRCLE

Ans. B

&A²

C²+D²-&AE%

'A

-C%

'B

-D


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EXAMPLE 36% What are the coorinates of center of the cur(e y² + x² - &y - 'x 31

a .-1/-10 .-'/-'0 c .1/'0 .'/10

CIRCLE

Ans. C

&A²

C²+D²-&AE%

'A

-C%

'B

-D


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EXAMPLE 37% Transfor4 into stanar equof the circle of

5eneral equation x² + y² - - 31 !

a .x-10² + .y-'0²31 .x-10

'0²32 c .x+10² + .y+'0²32 .x+.y+'0²31

CIRCLE

Ans. B

&A²

C²+D²-&AE%

'A

-C%

'A

-D

&A²

C²+D²-&AE%

'A

-C%

'B

-D


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EXAMPLE 38% What is the istance et#eecenters of the circles x² + y² + 'x + &y - 3 x² + y² - 6x - 2y + 7 !

a 7!7 777 c 6!7 767

CIRCLE

Ans. A

&A²

C²+D²-&AE%

'A

-C%

'B

-D


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EXAMPLE 39% The shortest istance fro4 .3the circle

x² + y² + &x ) 2y 1' is e

a '1 '3 c '* '7

CIRCLE

Ans. A

&A²

C²+D²-&AE%

'A

-C%

'B

-D


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EXAMPLE 40% The center of a circle is at .1one point on its circu4ference is .-1/-30 ,inother en of the ia4eter throu5h .-1/-30

a .'/&0 .3/*0

c .3/20 .1/30

CIRCLE

Ans. B


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EXAMPLE 41% A circle has its center at .-&/-is tan5ent to the line 3x + &y - * ! What raius of the circle8

a & * c 2 7

CIRCLE

Ans. B


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EXAMPLE 42% What is the raius of a circlethe follo#in5

equation x² + y² - 2x - &y - 1

a & 7 c * 2

CIRCLE

Ans. C

&A²

C²+D²-&AE%

'A

-C%

'B

-D


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EXAMPLE 43% The ia4eter of a circle escy

9x² + 9y² 12 is

a &3 129 c 63 &

CIRCLE

Ans. C

&A²

C²+D²-&AE%

'A

-C%

'B

-D


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EXAMPLE 44% Deter4ine the equation of the#hose raius is */ center on the line x' an

to the line 3x ) &y + 11 !

a .x-'0²+.y-'0²* .x-'.y+'0²'*

c .x-'0²+.y+'0²* .x-'0²'*

CIRCLE

Ans. B


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EXAMPLE 45% ,in the equation of the circlethe center at

.-&/-*0 an tan5ent to the lin7y - 1! !

a x²+y²+6x-1!y-1'!

x²+y²+6x-1!y+1'!c x²+y²+6x+1!y-1'! x

6x+1!y+1'!

CIRCLE

Ans. C


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EXAMPLE 46% ,in the (alue of the : for #hequation

x² + y² + &x ) 'y - : !represents a point circle

a * 2

c -2 -*

CIRCLE

Ans. A

&A²C²+D²-&AE

%

'A

-C%

'B

-D


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PARABOLA


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PARABOLA

Dist fro4 focus to(ertex a;atus rectu4 &aEccentricity 1

Ax² + Bx + Cy + D !

$eneral Equation%

PARABOLETT’s formula

&A-C

%

'A-B

%B²-&AD

&AC

hk a


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PARABOLA

Ax² + Bx + Cy + D

Ay² + By + Cx + D

.x-h0² &a.y-:0

.y-:0² &a.x-h0

a -a

-aa

Open Upward Open Downward

Open to the right Open to the left

&A-C

%

'A-B

%B²-&AD

&AC


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EXAMPLE 47% $i(en the equation of the par'y²+3x-&y+7 ! ;ocate the (ertex

a .*3/10 .*3

c .-*3/-10 .-*3

PARABOLA

Ans. D

&A-C

%

'A-B

%B²-&AD

&AC


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EXAMPLE 48% ,in the (ertex of the paraox²+6y-2x+17 !

a .3/10 .-3/-10 c .3/-10 .-3/10

PARABOLA

Ans. C

&A-C

%

'A-B

%B²-&AD

&AC


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EXAMPLE 49% A paraola has its axis parathe x-axis/ (ertex at .-1/70 an one en of t

latus rectu4 at .-1*&/3'0 ,in its equatio

a y²-11y-11x-2! ! y²11y+1&x-2! !

c y²-1&y+11x+2! ! <the ao(e

PARABOLA

Ans. C

&A-C

%

'A-B

%B²-&AD

&AC


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EXAMPLE 50% ,in the equation of the par#hose focus is at .1/!0 an (ertex is at .'/!

a y²+&x-6 ! y²+' c y²+2x-6 !


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EXAMPLE 51% The focus of the paraola y²is at

a .&/!0 .!/&0 c .!/-&0 .-&/!0

PARABOLA

Ans. A

&A-C

%

'A-B

%B²-&AD

&AC

PARABOLA


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EXAMPLE 52% What is the equation of theirectrix of the paraola y² + &x ) &y ) 6

a x 3 x ' c x 1 x &

PARABOLA

Ans. D

&A-C

%

'A-B

%B²-&AD

&AC

PARABOLA


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EXAMPLE 53% $i(en the paraola #ithhori=ontal axis that passes throu5h .!/!0 /

an .&/90

a '7x+11&y-1&y²! 11&y-1&y²!

c 17x+11&y+1&y²! '7x+119y-1&y²!

PARABOLA

Ans. A


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ELLIPSE

ELLIPSE


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ELLIPSE

a !(Horizontal Ellipse)

a " !(Vertical Ellipse)

E#$%$n&s '( ELLIPSE


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E#$%$n&s '( ELLIPSEy

,? ,@

c

a

? @C

irectrix irectrix

;atusrectu4

a

a - c

aeae

;atusrectu4

c

)*ESTIO+S ABO*T ELLIPSE


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)*ESTIO+S ABO*T ELLIPSE

D,s&an$ ('% ('/s &' &$ $&$ D,s&an$ ('% ('/s &' &$ C$n&$ D,s&an$ !$&$$n (', C'',na&$s '( &$ C$n&$ C'',na&$s '( &$ $&$ C'',na&$s '( &$ :', C'',na&$s '( &$ $n;',n&s '( &$

#a&/s $&/% L$n


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:ORM*LA

AB

ILLETT’s formula

; e < 1

ELLIPSE


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ELLIPSE

9x²+'*y²-''*!

Center!

"emi ma#or a!

"emi minor $!

ELLIPSE


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ELLIPSE

&x²+9y²+6x-3'!

Center!

"emi ma#or a!

"emi minor $!

ELLIPSE


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ELLIPSE

x²+&y²-'x-6y+1!

Center!

"emi ma#or a!

"emi minor $!

ELLIPSE


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ELLIPSE

Ans. B

%&'(

ELLIPSE


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EXAMPLE 55% $i(en the ellipse x²2& + y²Co4pute the eccentricity

a !** !79

c 11' !96

ELLIPSE

Ans. C

)&*+

ELLIPSE


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EXAMPLE 56% $i(en the ellipse x²2& + y²1 Co4pute the len5th of the latus rectu4

a & *

c 2 7

ELLIPSE

Ans. A

ELLIPSE


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EXAMPLE 57% $i(en the ellipse x²2& + y²1 What is the equation of the irectrix

a x 71* y

c x 71*y x

ELLIPSE

Ans. D

ELLIPSE


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EXAMPLE 58% $i(en the ellipse x²2& y²12 1

Co4pute the area oune ythe ellipse

a 3' 1!*3

c 1!3*3 13!*!

ELLIPSE

Ans. A

ELLIPSE


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EXAMPLE 59% $i(en the ellipse x²2& + y1

Co4pute the circu4feren

a 39&7 93&7 c 7&93 397&

Ans. D

ELLIPSE


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EXAMPLE 60% $i(en the ellipse'*x²+12y²-1*!x+1'6y+61!What is the coorinates of thecenter

a .3/-&0 .-3/&0

c .-&/30 .&/-30

Ans. A

'A-C

%'B-D

,

/ 0

ELLIPSE


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EXAMPLE 60% $i(en the ellipse'*x²+12y²-1*!x+1'6y+61!

What is the se4i 4aor an se4i4inor

a * an &

& an 3 c 2 an ' ' an *

Ans. A

-(./01

3

ELLIPSE


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What is the istance fro4 focus tothe center

a 1 '

c 3 &Ans. C

-(./01

3

ELLIPSE


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Co4pute the eccentricity

a !27 !*

c !33 !2 Ans. D

-(./01

ELLIPSE


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Co4pute the len5th of the latusrectu4

a *9

2& c 2' 6*

Ans. B

-(./01

3

LR

LR =

ELLIPSE


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What is the equation of theirectrices

a y&33 y-1'33 x336

&33 c y363 y&33 x636 -636

Ans. A

-(./01

3

ELLIPSE


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Co4pute the area oune y theellipse

a 36'2

'2 c 632' '!

Ans. D

3

ELLIPSE


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Co4pute the circu4ference

a &*'6 6'*&

c '6&* '6*&Ans. C

3

ELLIPSE


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EXAMPLE 67% An ellipse #ith an eccentrici!2* an has one of its foci ' units fro4 the

center The len5th of the latus rectu4 is neto

a 3* 36 c &' 3'

Ans. A

ELLIPSE


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EXAMPLE 68% An earth satellite has an apoof &!/!!!:4 an a peri5ee of 2/2!!:4

Assu4in5 the raius of the earth as 2/&!!:4#hat #ill e the eccentricity of the ellipticalpath escrie y the satellite #ith the centof the earth at one of the foci8

a !&2 !&9 c !*' !*2Ans. D

0).)) % % ) )

% 0 ) )

% 0 ) )


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HGPERBOLA

HGPERBOLA


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@E+ERAL E)*ATIO+ A-B>CD>E0

->(HorizontalHyperbola)

-(Vertical Hyperbola)

/

A-

B CD E

-

AB CD


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A-B>CD>E

0

-

AB

HGPERBOLA $n&$ a& F


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F

As,mptotes -,/51 4 -2/h1

$

a

6

&.h+c/:0

.h+a/:0?.h-a/:0

'-h.51

a7e

a

6

dire6tri6es

&.h-c/:0

6/a

a7e

)*ESTIO+S ABO*T HGPERBOLA


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D,s&an$ ('% (', &' &$ $&$ 2a D,s&an$ ('% (', &' &$ C$n&$ D,s&an$ !$&$$n (', 2

C'',na&$s '( &$ C$n&$ C'',na&$s '( &$ $&,$s C'',na&$s '( &$ :', C'',na&$s '( &$ $n;',n&s '( &$

#a&/s $&/% L$n $ 1 L$n


55/72

ILLETT’s formula

; e ( 1

8

8 A B

e)*at+on o the asymtote-y

HGPERBOLA

EXAMPLE 69 $i th h l 32 ² 2&


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EXAMPLE 69% $i(en the hyperola 32x²-2&'3!& Lo6ate the 6oordinates of the 6enter& a .!/!0 .6/20

c .2/60 .-6/20Ans. A

HGPERBOLA

EXAMPLE 69 $i th h l 32 ² 2&


57/72

EXAMPLE 69% $i(en the hyperola 32x²-2& '3!& 9ind a and $& a 6 an 2 3 an

c ' an 3 & anAns. A

HGPERBOLA

EXAMPLE 70 $i th h l 32 ²


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EXAMPLE 70% $i(en the hyperola 32x²-2&y² '3!& Lo6ate the 6oordinates of the:erti6es&

a .!/20 .!/-20 .2/!0 .-2/!0 c .!/60 .!/-60 .6/!0 .-6/!0

Ans. D

HGPERBOLA

EXAMPLE 71 $i(en the hyperola 32x²


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EXAMPLE 71% $i(en the hyperola 32x²-2&y² '3!& Lo6ate the 6oordinates of the fo6 a .!/20 .!/-20 .!/1!

.!/-1!0 c .2/!0 .-2/!0 .1!/! .-1!/!0

Ans. D

HGPERBOLA

EXAMPLE 72% $i(en the hyperola 32x²


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EXAMPLE 72% $i(en the hyperola 32x²-2&y² '3!& ;hat is the e66entri6it,& a 3& *2

c *& '*Ans. C

HGPERBOLA



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EXAMPLE 73% $i(en the hyperola 32x²-2&y² '3!& ;hat is the length of the latusre6tum&

a 7 9 c 6 2

Ans. B

HGPERBOLA



62/72

EXAMPLE 74% $i(en the hyperola 32x²-2&y² '3!& ;hat is the e


63/72

EXAMPLE 75% $i(en the hyperola 32x²-2& '3!& ;hat is the e


64/72

EXAMPLE 76% $i(en the hyperola 1'y²-&x²+7'y+12x+&& ! Lo6ate the 6enter& a .-'/30 .'/30

c .'/-30 .-'/-30Ans. C

HGPERBOLA

EXAMPLE 77% $i(en the hyperola 1'y²


65/72

EXAMPLE 77% $i(en the hyperola 1'y²-&x²+7'y+12x+&& ! 9ind the :alue of a and a aG' G1' a

G1' c a' 1' a 1'

Ans. B

HGPERBOLA

EXAMPLE 78% $i(en the hyperola 1'y²-


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EXAMPLE 78% $i(en the hyperola 1'y -&x²+7'y+12x+&& ! 9ind the trans:erse a2i a & 3

c ' 1Ans. A

HGPERBOLA


67/72

EXAMPLE 79% $i(en the hyperola 1'y²-&x²+7'y+12x+&& ! 9ind the 6on#ugate a2is

a & &G3 c ' 'G'Ans. B

HGPERBOLA



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EXAMPLE 80% $i(en the hyperola 1'y&x²+7'y+12x+&& ! 9ind the 6oordinates offo6i&

a .'/10 .'/-70 .'.-'/70 c .'/-10 .'/-70 .'.'/70

Ans. A

HGPERBOLA



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EXAMPLE 81% $i(en the hyperola 1'y&x²+7'y+12x+&& ! 9ind the e66entri6it,& a * 3

c & 'Ans. D

HGPERBOLA



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EXAMPLE 81% $i(en the hyperola 1'y&x²+7'y+12x+&& ! 9ind the length of latus a 1! 6

c & 1'Ans. D

HGPERBOLA



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8 $ e e ype o a y&x²+7'y+12x+&& ! 9ind the e


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yp y&x²+7'y+12x+&& ! 9ind the e

mathematics ge review session 5

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