ParabolaHard
Question
The co-ordinates of a point on the parabola 2y = x2 which is nearest to the point (0, 3) is
Options
A.(2, 2)
B.(-√2, 1)
C.(√2, 1)
D.(- 2, 2)
Solution
2y = x2
2y′ = 2x
y′ = h
Equation of normal at (h, k)
(y - k) = - 1/h (x - h)
As it passes through (0, 3)
So, (3 - k) h = - (-h)
⇒ (3 - k) h = h
or, h (3 - k - 1) = 0
or, h (2 - k) = 0
or, 2h - hk = 0
or, 2h -
= 0 
or, 4h - h3 = 0
or, h = 0,
Required points are (2, 2) & (-2, 2)
∴ (Rejecting (0, 0) since, its distance from point (0,3) is 3 which is not shortest.)
2y′ = 2x
y′ = h
Equation of normal at (h, k)
(y - k) = - 1/h (x - h)
As it passes through (0, 3)
So, (3 - k) h = - (-h)
⇒ (3 - k) h = h
or, h (3 - k - 1) = 0
or, h (2 - k) = 0
or, 2h - hk = 0
or, 2h -
= 0 or, 4h - h3 = 0
or, h = 0,
Required points are (2, 2) & (-2, 2)
∴ (Rejecting (0, 0) since, its distance from point (0,3) is 3 which is not shortest.)
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