wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Consider two concentric circles C1:x2+y24=0 and C2:x2+y29=0. A parabola is drawn through the points where C1 meets yaxis and having an arbitrary tangent of C2 as its directrix. If C is the curve of locus of focus of drawn parabola and e is the eccentricity of the curve C, then the value of 12e is

Open in App
Solution

Let P(h,k) be the focus of parabola.
Circle C1 meets yaxis at (0,±2)
An arbitrary tangent to C2 is xcosθ+ysinθ=3
Parabola passes through (0,±2) and its directrix is
xcosθ+ysinθ3=0

We know that in a parabola, distance between a point and the directrix and the distance between focus and the point are equal.
h2+(k2)2=(2sinθ3)2 (1)
and h2+(k+2)2=(2sinθ+3)2 (2)
From equations (1) and (2),
k3=sinθ, h5=cosθ
Since sin2θ+cos2θ=1
k2(3)2+h2(5)2=1
Required locus is x25+y29=1
Eccentricity, e=159
e=23
12e=8

flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Line and a Parabola
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon