Conic Sections Question 86
Question: The two parabolas $ y^{2}=4x $ and $ x^{2}=4y $ intersect at a point P, whose abscissa is not zero, such that
Options:
A) They both touch each other at P
B) They cut at right angles through P
C) The tangents to each curve at P make equal angles with the x-axis
D) None of these
Show Answer
Answer:
Correct Answer: C
Solution:
Solving $ x^{2}=4y $ and $ y^{2}=4x, $ we get $ x=0,y=0 $ and $ x=4,y=4 $ . Therefore the co-ordinates of P are (4,4). The equations of the tangents to the two parabolas at (4,4) are $ x+y-8=0 $ and $ x-y=0 $
…..(i) $ x-2y+4=0 $ ……(ii) Now, $ m_1= $ Slope of (i) $ =-\frac{1}{2}, $
$ m_2= $ Slope of (ii) $ =\frac{1}{2} $
$ m_1m_2=1\text{ i.e., }\tan {\theta_1}\tan {\theta_2}=1 $
BETA
