Conic Sections Question 187
Question: Let a circle tangent to the directrix of a parabola $ y^{2}=2ax $ has its centre coinciding with the focus of the parabola. Then the point of intersection of the parabola and circle is
[Orissa JEE 2005]
Options:
A) (a, -a)
B) $ (a/2,\ a/2) $
C) $ (a/2,\ \pm a) $
D) $ (\pm a,\ a/2) $
Show Answer
Answer:
Correct Answer: C
Solution:
Given parabola is $ y^{2}=2ax $
Focus (a/2, 0) and directrix is given by $ x=-a/2 $ , as circle touches the directrix. Radius of circle = distance from the point (a/2, 0) to the line $ (x=-a/2) $
$ =\frac{| \frac{a}{2}+\frac{a}{2} |}{\sqrt{1}}=a $
Equation of circle be $ {{( x-\frac{a}{2} )}^{2}}+y^{2}=a^{2} $
…..(i) also $ y^{2}=2ax $ ……(ii) Solving (i) and (ii) we get $ x=\frac{a}{2},\ -\frac{3a}{2} $
Putting these values in $ y^{2}=2ax $ we get $ y=\pm a $ and $ x=-3a/2 $ gives imaginary values of y. Required points are $ (a/2,\ \pm a) $ .
BETA
