Conic Sections Question 100
Question: If $ lx+my+n=0 $ is tangent to the parabola $ x^{2}=y $ , then condition of tangency is
[RPET 1999]
Options:
A) $ l^{2}=2mn $
B) $ l=4m^{2}n^{2} $
C) $ m^{2}=4ln $
D) $ l^{2}=4mn $
Show Answer
Answer:
Correct Answer: D
Solution:
Given that $ lx+my+n+0 $ ……(i) $ x^{2}=y $ ……(ii) The point of intersection of the line and parabola are obtained by solving (i) and (ii) simultaneously substituting the values of $ x $ from (i) in (ii), we get $ {{( \frac{my+n}{l} )}^{2}}=y $
$ \Rightarrow $ $ m^{2}y^{2}+n^{2}+2mny=yl^{2} $
therefore $ m^{2}y^{2}+(2mn-l^{2})y+n^{2}=0 $ …..(iii)
If lines (iii) touches the parabola (ii), then discriminant = 0
therefore $ {{(2mn-l^{2})}^{2}}=4m^{2}n^{2} $
$ \Rightarrow 4m^{2}n^{2}+l^{4}-4mnl^{2}=4m^{2}n^{2} $
therefore $ l^{2}=4mn $ .
BETA
