* These equations represent mirror lines
to their plus counterparts. Both lines are parallel.
* In fact, x/b ± y/a =1 and y = mx ± c1
are various forms of the equation of a st. line. Which form one
chooses is a matter of convention. We have chosen the most used
format. * The equation ax+by+c=0 represents a
st.line whose xintercept is b, yintercept is a and c represents
the negative of the products of the intercepts i.e. c=ab. Hence
ay+bx+c=0 represents a st. line whose intercepts have been
exchanged. *The equation of a straight line which
is drawn perpendicular to the line given by equation ax+by+c=0 and
passes through the point of interception(0,c/b) or (0,a) of the
line with yaxis is given by b^{2}xabyac=0
or b^{2}xaby+a^{2}b=0.
*The equation of a straight line which is drawn perpendicular to the
line given by equation ax+by+c=0 and passes through the point of
interception(c/a,0) or (b,0) of the line with xaxis is given by
a^{2}yabxbc=0 or or a^{2}yabx+ab^{2}=0. * the perpendicular distance from a point
(x1,y1) to a st.line with equation ax + by + c =0
is given by d= (ax1+by1+c) / √ (a^{2} + b^{2} )
*In A general equation of the type
Ax^{2} +Bxy+Cy^{2} +Dx +Ey+F=0,
the eccentricity e is given by e = √(2 √[ (AC)^{2} + B^{2} ])
/√ (n(A+C) +√[ (AC)^{2} + B^{2} ] )
& n =1 if (ACB^{2}/4) < 0 and n = 1 if (ACB^{2}/4) > 0
Pl. note that D,E,F are not contributing to eccentricity. If B^{2}
=4AC, curve is a parabola. The coefficient B contributes to
eccentricity and solely contributes to the slanting of the
curve with respect to the coordinate axis. Since B^{2 }
is positive, its effect is to increase the eccentricity e whereas
the role of second factor AC depends on overall sign. Taking A'=2A
and C'=2C. we can write B^{2}
= > <A'C' or B= > <√(A'C'). Let us state that M=√(A'C')= geometric
mean of A',C'. Hence B may be equal to, less than or greater than
the geometric mean. In case of nonequality, one can calculate the
standard deviation.
* In Ax^{2} +Bxy+Cy^{2} +Dx +Ey+F=0
if B^{2}4AC < 0 ellipse
if B^{2}4AC < 0 , A=C, B=0 , circle
if B^{2}4AC = 0 , parabola
if B^{2}4AC > 0 , hyperbola
if B^{2}4AC > 0, A+C=0, Rectangular hyperbola
^ a1x1 +b1y1+c1=0 ; coordinate
of yintercept (0,a1). length from this point to (x1.y1) is (x10)^{2}
+(y1a1)^{2} = L^{2}
Let the st.line perpendicular to the above line, having a point
(x2,y2) which is analogous to (x1,y1) have the equation
a2x2+b2y2+c2=0 .......(II)
and length from this point to (x2.y2) to (0,a1) is
(x20)^{2}
+(y2a1)^{2} = n1=L^{2} ......(I)
Squaring & rearranging both sides of .....(II), we get
a^{2}2x^{2}2b^{2}2y^{2}2c^{2}22b2c2y2=0
.......(III)
multiplying both sides of (I) by
a^{2}2 , we get
a^{2}2x^{2}2+a^{2}2y^{2}22a^{2}2a1y2+a^{2}2a^{2}1n2=0
.......(IV) where n2=a^{2}2 *n1
(IV)  (III) = y^{2}2(a^{2}2+b^{2}2)
+2y2(b2c2a^{2}2a1 ) +(c^{2}2+a^{2}2a^{2}1
n2)=0
Taking A= (a^{2}2+b^{2}2),
B= 2(b2c2a^{2}2a1 ), C=(c^{2}2+a^{2}2a^{2}1
n2)
this is a quadratic equn A y^{2}2
+By2 +C=0
y2=B ± √( B^{2} 4AC) /2A
x2= (C2b2y2)/a2 ;
when the rotation is through the xintercept point, the equn becomes
(IVa)  (IIIa) = x^{2}2(a^{2}2+b^{2}2)
+2x2(a2c2b^{2}2b1 ) +(c^{2}2+b^{2}2b^{2}1
n'2)=0
where
n'2=b^{2}2 *n'1
and n'= (x2b1)^{2}
+(y2)^{2} = L^{2}
and
A= (a^{2}2+b^{2}2),
B= 2(a2c2b^{2}2b1 ), C=(c^{2}2+b^{2}2b^{2}1
n'2)
this is a quadratic equn A x^{2}2
+Bx2 +C=0
x2=B ± √(B^{2}4AC) / 2A
y2= (C2a2x2)/b2 ;
