mathematics ge review session 5
TRANSCRIPT
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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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EXAMPLE 61% $i(en the ellipse'*x²+12y²-1*!x+1'6y+61!
What is the istance fro4 focus tothe center
a 1 '
c 3 &Ans. C
-(./01
3
ELLIPSE
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EXAMPLE 62% $i(en the ellipse'*x²+12y²-1*!x+1'6y+61!
Co4pute the eccentricity
a !27 !*
c !33 !2 Ans. D
-(./01
ELLIPSE
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EXAMPLE 63% $i(en the ellipse'*x²+12y²-1*!x+1'6y+61!
Co4pute the len5th of the latusrectu4
a *9
2& c 2' 6*
Ans. B
-(./01
3
LR
LR =
ELLIPSE
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EXAMPLE 64% $i(en the ellipse'*x²+12y²-1*!x+1'6y+61!
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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EXAMPLE 65% $i(en the ellipse'*x²+12y²-1*!x+1'6y+61!
Co4pute the area oune y theellipse
a 36'2
'2 c 632' '!
Ans. D
3
ELLIPSE
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EXAMPLE 66% $i(en the ellipse'*x²+12y²-1*!x+1'6y+61!
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
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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&
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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
EXAMPLE 73% $i(en the hyperola 32x²
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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
EXAMPLE 74% $i(en the hyperola 32x²
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EXAMPLE 74% $i(en the hyperola 32x²-2&y² '3!& ;hat is the e
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EXAMPLE 75% $i(en the hyperola 32x²-2& '3!& ;hat is the e
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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
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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
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EXAMPLE 79% $i(en the hyperola 1'y²-&x²+7'y+12x+&& ! 9ind the 6on#ugate a2is
a & &G3 c ' 'G'Ans. B
HGPERBOLA
EXAMPLE 80% $i(en the hyperola 1'y²-
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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
EXAMPLE 81% $i(en the hyperola 1'y²-
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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
EXAMPLE 81% $i(en the hyperola 1'y²-
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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
EXAMPLE 81% $i(en the hyperola 1'y²-
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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