Coordinates of the point on the straight line x+y=4 which is the nearest to the parabola y2=4(x-10) is
Answers
Given : parabola y²=4(x-10) & line x + y = 4
To Find : Coordinates of the point on the straight line x+y=4 which is the nearest to the parabola
Solution:
y² = 4(x - 10)
=> 2y dy/dx = 4
=> dy/dx = 2/y
Slope of line x + y = 4 is
-1 ( as y = -x + 4)
Hence slope of Tangent at parabola = slope of line x + y = 4
(as shortest distance would be perpendicular )
=> dy/dx = 2/y = -1
=> y = -2
(-2)² = 4(x - 10)
=> 1 = x - 10
=> x = 11
( 11 , -2 ) is the point
( 11 , -2 ) is the point on parabola
Slope of perpendicular line = 1 ( line of shortest distance )
=> y - (-2) = 1 (x - 11)
=> y + 2 = x - 11
=> x - y = 13
x - y = 13
x + y = 4
=> 2x = 17 => x = 17/2
=> y = -9/2
Point ( 17/2 , - 9/2) is nearest to the parabola y² = 4(x - 10) on line x + y = 4
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