Question

The set of values of a for which

$(13x - 1)^{2} + (13y - 2)^{2} = a(5x + 12y - 1)^{2}$represents an ellipse, is

• A. 1 < a < 2
• B. 0 < a< 1
• C. 2 < a < 3
• D. None

Answer 1

Answer: (B)

0 < a< 1

Answer 2

We have

$(13x - 1)^{2} + (13y - 2)^{2} = a(5x + 12y - 1)^{2}$ …….(1)

$\because$ L.H.S. is +ve

$\therefore$ R.H.S is also +ve then a > 0 …(2)

Again equation (1) can be written as

$(169 - 25a)x^{2} + (169 - 144a)y^{2} - 120axy + (...)x + (...)y + (5 - a) = 0$It is an ellipse.

$\therefore H^{2} < AB$

then $3600a^{2} < (169 - 25a)(169 - 144a)$

⇒ a < 1 ………………..(3)

from (2) and (3) we get

$0 < a < 1$