Let y = a 
+ bx be curve, (2x – y) + λ (2x + y – 4) = 0
be family of lines.
(i) If curve has slope – 
at (9, 0) then a tangent belonging to family of lines is –
Text Solution
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Ans.
(i)
Sol. 0 = 3a + 9b, – 
= 
+ b
a = – 3b, – 
= – 
+ b– 
= 
b = – 1
⇒ a = 3 ; y = 3 
– x
Point (1, 2) lies on curve.
= 
– 1
at (1, 2) is 
y – 2 = 
(x – 1) ⇒ x – 2y + 3 = 0
(ii)
Sol. sin θ = 
PA = 2 cosec θ
cos θ = 
BP = sec θ
AB = AP + BP = 2 cosec θ + cos θ
Min. value of AB = (2 2/3 +1) 3/2
(iii)
Sol. Let y – 2 = m (x – 1) be a chord.
y = mx + 2 – m
(mx + 2 – m)
2 – 4x – 4 (mx + 2 – m) + 4 = 0
i.e. (mx – m) 2 – 4x = 0
m 2 x 2 – (2m 2 + 4) x + m 2 = 0
x 1 + x 2 = 
; x 1 x 2 = 1
|x 2 – x 1 | = 
= 
|x 2 – x 1 | = 
Length of diagonal = 
= 
(m 2 + 1)
Length of diagonal of perpendicular chord
= 4 m 2 
= 4 (1 + m 2 )
Area = 
(1 + m 2 ) 4 (1 + m 2 ) = 8 (m 2 + 2 + 
)
⇒ Minimum area =16 + 8 × 2 = 32.
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