For ellipse 
+ 
= 1 (a > b) match the following.
Column-I | Column-II |
(i) Locus of point of intersection of two perpendicular tangents | [A] x2 + y2 = a2 |
(ii) Locus of foot of perpendicular from focus on any tangent | [Β] (x2 + y2)2 = a2x2 + b2y2 |
(iii) Locus of foot of perpendicular from centre on any tangent | [C]4(x2 + y2)2 = a2x2 + b2y2 |
(iv) Locus of mid point of segment OM where M is foot of perpendicular from centre O to any tangent | [D] x2 + y2 = a2 + b2 |
Text Solution
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Ans.
(i) [D]
(ii) [A]
(iii) [B]
(iv) [R]
Sol.
(i) Locus is director circle.
(ii) Locus is auxiliary circle.
(iii) Equation of tangent is
y = mx + 
… (1)
and equation of perpendicular from centre is
y = 
… (2)
Eliminate m from (1) and (2)
(iv) Let M is (x 1 , y 1 ) and mid point of OM is (h, k)
∴ h = 
, k = 
x 1 = 2h, y 1 = 2k
∴ (x 1 , y 1 ) satisfy equation (1)
∴ 2 β = m (2 α ) + 
… (3)
Also (x 1 , y 1 ) satisfy equation (2)
∴ 2 β = 
… (4)
Eliminate m from (3) and (4)
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