Let f be a strictly decreasing function defined on R such that f(x)>0,∀x∈R. Let \(\frac{x^2}{f(a^2+5a+3)}+\frac{y^2}{f(a+15)}=1\) be an ellipse with a major axis along the y-axis. The value of 'a' can lie in the interval(s)
- (-∞,-6)
- (-6,2)
- (2,∞)
- (-∞,∞)
The Correct Option is A, C
Solution and Explanation
The given equation represents an ellipse centered at the origin, with its major axis along the y-axis. The standard form equation of such an ellipse is:
x2/b2+y2+a2=1
where a is the length of the semi-major axis and b is the length of the semi-minor axis.
Comparing the given equation (2+5+3)2+(+15)2=1f(a2+5a+3)x2+f(a+15)y2=1 to the standard form, we see that 2+5+3a2+5a+3 corresponds to a2 and a+15a corresponds to b2.
Given that f is a strictly decreasing function and f(x)>0 for all x, we can conclude that a2+5a+3 must be greater than a+15 for the major axis to be along the y-axis. This leads to the inequality 2+4−12>0a2+4a−12>0, which factors as (2)(6)>0(a−2)(a+6)>0.
This implies that a must lie in the interval (−∞,−6)∪(2,∞)(−∞,−6)∪(2,∞) for the major axis to be along the y-axis and for the given equation to represent an ellipse. Hence, the intervals (−∞,−6)(−∞,−6) and (2,∞)(2,∞) are the correct intervals for the value of a.
The correct answer is/are option(s):
(A): (-∞,-6)
(C): (2,∞)
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Concepts Used:
Functions
A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Let A & B be any two non-empty sets, mapping from A to B will be a function only when every element in set A has one end only one image in set B.
Kinds of Functions
The different types of functions are -
One to One Function: When elements of set A have a separate component of set B, we can determine that it is a one-to-one function. Besides, you can also call it injective.
Many to One Function: As the name suggests, here more than two elements in set A are mapped with one element in set B.
Moreover, if it happens that all the elements in set B have pre-images in set A, it is called an onto function or surjective function.
Also, if a function is both one-to-one and onto function, it is known as a bijective. This means, that all the elements of A are mapped with separate elements in B, and A holds a pre-image of elements of B.
Read More: Relations and Functions
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