Maths-
General
Easy
Question
Statement‐I The ellipse 
and 
are congruent.
Statement‐II The ellipse and have the same eccentricity.
- Statement‐I is True, Statement‐II is True; Statement‐II is a correct explanation for Statement‐I.
- Statement‐I is True, Statement‐II is True; Statement‐II is NOT a correct explanation for Statement‐I
- Statement‐I is True, Statement‐II is False
- Statement‐I is False, Statement‐II is True
Hint:
We are given equation of two ellipses.There are two statements related to this ellipses. In first statement we have to show if the given ellipses are congruent. Congruent figures have same size and same shape. In the second statement we have to find their eccentricity and compare it.
The correct answer is: Statement‐I is True, Statement‐II is True; Statement‐II is NOT a correct explanation for Statement‐I
We are given two ellipse.
Let the first ellipse be denoted by A and second be denoted by B
The equation of ellipse A is
The equation of ellipse B is
The general equation of ellipse is 
If a > b then, a is the semi-major axis and b is the semi minor axis of the ellipse.
Similarly, if b > a then b is the semi-major axis and a is the semi minor axis.
We will check the statements one by one.
Statement I: we have to find if the ellipse are congruent.
Condition for two ellipses to be congruent : The ellipses are said to be congruent if their semi-major axis and semi-minor axis are equal.
For ellipse A
Semi-major axis is b2 = 16
b = 4
Semi-minor axis is a2 = 9
a = 3
For ellipse B
Semi-major axis is a2 = 16
a = 4
Semi-minor axis is b2 = 9
b = 3
As both the semi-major axis and semi-minor axis of ellipse A and B are equal, both the ellipse are congruent.
Statement I is true.
Statement II: We have to find if both the ellipse have same eccentricity.
Eccentricity of an ellipse is the ratio of the distance of foci from the center and the distance of one end of vertices of the ellipse from the center.
Let us denote the distance from the center be denoted by c. And let the vertices be the semi-major axis. Let eccentricity of ellipse A be eA and ellipse B be eB.
So, the formula to find the distance from the foci is as follows:
If a is the semi-major axis and b is the semi-minor axis then 
For ellipse A
For ellipse B
c = 
So, the ecentricity of both the ellipses are same
Statement II is true.
Both the statements are true. Statement I needs the major and minor axis to be equal. It doesn't depend on statement II.
So, both the statements are true. But statement II is not the correct explanation of statement I.
For such questions, we should know properties of ellipse. We should know all the formulas related to ellipse. The axis which is larger is always the major axis.
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