Prove that the Length of the Latus Rectum of the Hyperbola :
Here we are going to see how to prove the length of latus rectum of the hyperbola as 2b2/a
Question 1 :
Prove that the length of the latus rectum of the hyperbola (x2/a2) - (y2/b2) = 1 is 2b2/a.
In the picture given above LSL' is the latus rectum and LS is called semi latus rectum TS'T' is also a latus rectum.
The coordinates of L are (ae, SL)
As L lies on the hyperbola.
(x2/a2) - (y2/b2) = 1
The coordinate will satisfy the equation of the hyperbola
((ae)2/a2) - ((SL)2/b2) = 1
(a2e2/a2) - ((SL)2/b2) = 1
e2 - 1 = ((SL)2/b2)
(SL)2 = b2(e2 - 1) -----(1)
b2 = a2 (e2 - 1)
(e2 - 1) = b2/a2
By applying the value of (e2 - 1) in (1), we get
(SL)2 = b2(b2/a2)
(SL)2 = b4/a2
SL = b2/a (length of semi latus rectum)
SL + SL' = 2b2/a
Question 2 :
Show that the absolute value of difference of the focal distances of any point P on the hyperbola is the length of its transverse axis
Let p(x,y) be any point on the hyperbola
(x2/a2) - (y2/b2) = 1.a
Let MPM' be the perpendicular through P on directrices ZK and Z'K'. Now by definition we get,
SP = e ∙ PM
SP = e ∙ NK
SP = e (CN - CK)
SP = e(X - (a/e))
SP = eX - a
S'P = e ∙ PM'
⇒ S'P = e ∙ (NK')
⇒ S'P = e (CK' + CN)
⇒ S'P = e(X + (a/e))
S'P = eX + a
S'P - SP = (a + ex) - (ex - a)
= a + ex - ex + a
= length of transverse axis.
After having gone through the stuff given above, we hope that the students would have understood, "Prove that the Length of the Latus Rectum of the Hyperbola".
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
APTITUDE TESTS ONLINE
ACT MATH ONLINE TEST
TRANSFORMATIONS OF FUNCTIONS
ORDER OF OPERATIONS
Decimal place value worksheets
Area and perimeter
Different forms equations of straight lines
MATH FOR KIDS
HCF and LCM word problems
Word problems on quadratic equations
Word problems on comparing rates
Ratio and proportion word problems
Converting repeating decimals in to fractions