Two tangents PA and PB are drawn from a point P(h, k) to the ellipse E : 
(a > b). Angle of the tangents with the positive x - axis are θ 1 and θ 2 . Normals at A and B are intersecting at Q point.On the basis of above information answer the following questions.
(i) Locus of P, if tan θ 1 . tan θ 2 = 4, is
= 2(x + a) mid point of AB 2xy = λ (y 2 – b 2 )
= 
orthocentre of Δ PAB xy = λ
(ii) Circumcentre of Δ QAB is can't say (iii) Locus of P, if cot θ 1 + cot θ 2 = λ , is x 2 + xy = λ
Text Solution
Verified by ExpertsA
(i) y = mx ± 
k = mh ± 
(k–mh) 2 = a 2 m 2 + b 2
m 2 (h 2 – a 2 ) – 2mhk + k 2 – b 2 = 0 (m 1 = tan θ 1 , m 2 = tan θ 2 )
m 1 m 2 = 
⇒ 
⇒ 
(ii) ∠ QAP = ∠ PBQ = 90º
hence a circle drawn taking 'PQ' as diameter will pass through B,A,P,Q
∴ center will be mid point of PQ
(iii) m 1 + m 2 = 
and cot θ 1 + cot θ 2 = λ
⇒ 
⇒ 
= λ ⇒ 
⇒ 2hk = λ (k 2 – b 2 )
2xy = λ (y 2 – b 2 )
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