Question
Question: If \({e_1}\) and \({e_2}\) are respectively the eccentricities of the ellipse \(\dfrac{{{x^2}}}{{18}...
If e1 and e2 are respectively the eccentricities of the ellipse 18x2+4y2=1 and the hyperbola 9x2−4y2=1, then the relation between e1 and e2 is:
A. 3(e1)2+(e2)2=2
B. (e1)2+2(e2)2=3
C. 2(e1)2+(e2)2=3
D. (e1)2+3(e2)2=2
Solution
Here, we will be using the formulas for the eccentricities of any general ellipse a2x2+b2y2=1 where a>b and any general hyperbola a2x2−b2y2=1 which are e1=1−(a2b2) and e2=1+(a2b2) respectively.
Complete step-by-step answer:
Given equation of ellipse is 18x2+4y2=1⇒(32)2x2+22y2=1 →(1)
and equation of hyperbola is 9x2−4y2=1⇒32x2−22y2=1 →(2)
As we know the formula for eccentricity of any general ellipse a2x2+b2y2=1 →(3) where a>b is given by
e1=1−(a2b2) →(4)
On comparing equations (1) and (3), we get
a=32 and b=2
Clearly, 32>2 i.e., b>a so the formula given by equation (4) is valid in this case.
Putting the above values in equation (4), we get
e1=1−((32)222) = 1−(184)=1818−4=1814=97
⇒(e1)2=97 →(5)
Also we know the formula for eccentricity of any general hyperbola a2x2−b2y2=1 →(6) is given by
e2=1+(a2b2) →(7)
On comparing equations (2) and (6), we get
a=3 and b=2
Putting the above values in equation (7), we get
e2=1+(3222) = 1+(94)=99+4=913
⇒(e2)2=913 →(8)
Now, we will use the equations (5) and (8) to verify which one of the relations given in the options is correct.
Now, taking LHS of option A i.e., 3(e1)2+(e2)2=3×(97)+913=921+913=934=2=RHS
Now, taking LHS of option B i.e., (e1)2+2(e2)2=97+2×(913)=97+926=933=311=3=RHS
Now, taking LHS of option C i.e., 2(e1)2+(e2)2=2×(97)+913=914+913=927=3=RHS
Now, taking LHS of option B i.e., (e1)2+3(e2)2=97+3×(913)=97+939=946=2=RHS
Clearly only option C is satisfied so the required relation is 2(e1)2+(e2)2=3.
Hence, option C is correct.
Note: In these types of problems using the general equation for any ellipse or hyperbola, values of various parameters used in the formulas for eccentricities of the ellipse and hyperbola are evaluated. After that with the help of options given, the relation which satisfies the evaluated values of the eccentricities is the required relation between the eccentricities of the given ellipse and hyperbola.