List-I List-IIA) The number of rational points on the ellipse x29+y24=1 1) 4B) The number of integral points on the ellipse x29+y24=1 2) 2 C) The number of integral points on the ellipse x23+y21=1 3) ∞The correct matching is

x29+y24=1 y24=1-x29>0 ⇒x2-9 0 ⇒y2-4 0 ⇒x2-3 0 ⇒y2-1<0 ⇒-1<y<1no of integral points=2

List-I List-IIA) The number of rational points on the ellipse x29+y24=1 1) 4B) The number of integral points on the ellipse x29+y24=1 2) 2 C) The number of integral points on the ellipse x23+y21=1 3) ∞The correct matching is

x29+y24=1 y24=1-x29>0 ⇒x2-9 0 ⇒y2-4 0 ⇒x2-3 0 ⇒y2-1<0 ⇒-1<y<1no of integral points=2

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List-I List-II
A) The number of rational points on the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ 1) 4
B) The number of integral points on the ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$ 2) 2
C) The number of integral points on the ellipse $\frac{x^{2}}{3} + \frac{y^{2}}{1} = 1$ 3) $\infty$
The correct matching is

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A B C
1 2 3

A B C
2 3 1

A B C
3 1 2

A B C
3 2 1

answer is B.

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Detailed Solution

$\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1 \frac{y^{2}}{4} = 1 - \frac{x^{2}}{9} > 0 \Rightarrow x^{2} - 9 < 0 \Rightarrow - 3 < x < 3 \frac{x^{2}}{9} = 1 - \frac{y^{2}}{4} > 0 \Rightarrow y^{2} - 4 < 0 \Rightarrow - 2 < y < 2$

no of rational points $= \infty$

no of integral points=4

$\frac{x^{2}}{3} + \frac{y^{2}}{1} = 1 y^{2} = 1 - \frac{x^{2}}{3} > 0 \Rightarrow x^{2} - 3 < 0 \Rightarrow - \sqrt{3} < x < \sqrt{3} \frac{x^{2}}{3} = 1 - y^{2} > 0 \Rightarrow y^{2} - 1 < 0 \Rightarrow - 1 < y < 1$