Match the column:
Column-I | Column-II |
(i) Normals are drawn to the parabola y2 = 4x at P (t2, 2t); where t < 0, which meets the curve again at Q. If ordinate of Q is minimum then PQ2 is | [A] 3 |
(ii) From point on circle x2 + y2 = 36, tangents are drawn to hyperbola x2 – y2 = 36, locus of mid points of chords of contact is(x2 – y2)2 = 12λ (x2 + y2) then λ = | [B]0 |
(iii) Maximum length of chord of ellipse + y2 = 1, such that eccentric angles of its extremities differ by π/2 is | [C]108 |
(iv) Three distinct lines are drawn in a plane & there are exactly n circles in plane tangent to all three lines, then n can be | [D]2 |
[E]4 |
Text Solution
Verified by Experts(i) [C]; (ii) [A]; (iii) [D]; (iv) [B]
Ans.
(i) [C]
(ii) [A]
(iii) [D]
(iv) [B],[D],[E]
Sol.
(i) Point P is (t 2 , 2t)
& Q is 
∴ – t 
≥ 
for minimum, t = 
∴ Points are P (2, –2 
) & Q (8, 
)
∴ PQ 2 = 108
(ii) Equation of chord of contact is
T = 0
x cos θ – y sin θ = 6 ……. (i)
Let M (h, k) is mid point of AB
∴ Equation of AB is T = S 1
⇒ hx – ky = h 2 –k 2 …….(ii)
∴ comparing (i) & (ii)
cos θ = 
sin θ = 
square & add, locus is
(x 2 – y 2 ) 2 = 36 (x 2 + y 2 )
∴ λ = 3
(iii) Let P ( 
cos θ , sin θ ) & Q (– 
sin θ , cos θ )
are extremities of chord
∴ Length PQ = 
≤ 2
∴ ( PQ) max. = 2
(iv) case - (i) : If lines form a triangle then n = 4
i.e. 3 excircle & 1 incircle
(ii) Two lines are parallel & third intersect them then n = 2
(iii) All lines are parallel or lines are concurrent there n = 0
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