Conic-Sections Question 597
Question: Let E be the ellipse $ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $ and C be the circle $ x^{2}+y^{2}=9. $ Let $ P=(1,2) $ and $ Q=(2,1) $ Which one of the following is correct?
Options:
A) Q lies inside C but outside E
B) Q lies outside both C and E
C) P lies inside both C and E
D) P lies inside both C but outside E.
Show Answer
Answer:
Correct Answer: D
Solution:
[d] Given equation of ellipse E is $ \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 $
$ \Rightarrow \frac{4x^{2}+9y^{2}}{36}=1\Rightarrow 4x^{2}+9y^{2}=36 $
$ \Rightarrow 4x^{2}+9y^{2}-36=0 $ ?. (1) And C: Eqs of circle is $ x^{2}+y^{2}=9 $ Which can be rewritten as $ x^{2}+y^{2}-9=0 $ ? (2) For a point P (1, 2) we have $ 4{{(1)}^{2}}+9{{(2)}^{2}}-36=40-36>0 $ [from (1)] and $ 1^{2}+2^{2}-9=5-9<0 $ [from (2)]
$ \therefore $ Point P lies outside of E and inside of C.
BETA
