Conic Sections Question 213
Question: The line $ y=mx+c $ intersects the circle $ x^{2}+y^{2}=r^{2} $ at the two real distinct points if
Options:
A) $ -r\sqrt{1+m^{2}}<c<r\sqrt{1+m^{2}} $
B) $ -r<c<r $
C) $ -r\sqrt{1-m^{2}}<c<r\sqrt{1+m^{2}} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] Given line is $ y=mx+c $
- (1) and the given circle is $ x^{2}+y^{2}=r^{2} $ - (2) Solving (1) and (2), we get $ (1+m^{2})x^{2}+2mcx+c^{2}-r^{2}=0 $
…… (3) For two real distinct points of intersection, both the roots of (3) must be real distinct.
$ \therefore 4m^{2}c^{2}-4(1+m^{2})(c^{2}-r^{2})>0 $
$ \Rightarrow c^{2}<r^{2}(1+m^{2})\Rightarrow $
$ -r\sqrt{1+m^{2}}<c<\sqrt{1+m^{2}} $
BETA
