Question

The shortest distance between the line
y = x and the curve y2 = x – 2 is​

Answer 1

Answer:

i don t know the answer so sorry but you mark me as a brainlist

Answer 2

Since solution to y = x and y^2 = x - 2 doesn't exist, the shortest distance between them must be perpendicular to the curve and line. It means, both would have same slope at such condition.

For slope of y^2 = x - 2, diff. wrt. x :

=> 2yy' = 1 => y' = 1/(2y)

For slope of line y = x, use y = mx + c

=> m = 1

As both should have same slope, m = y'

=> 1 = 1/2y => y = 1/2

Therefore, when y = 1/2 (on the curve)

(1/2)^2 = x - 2 => 9/4 = x

Hence the point where perpendicular meets on curve is (9/4, 1/2).

Now we need to find the point on line that is perpendicular to curve(or line with (9/4, 1/2) and slope = - 1/1).

Eqⁿ of perpendicular is

=> y - 1/2 = -1 (x - 9/4)

=> 4y + 4x - 11 = 0

Now, the lines y = x and 4y + 4x - 11 = 0 meet at :

(x, y) = (11/8, 11/8).

We have a point on line(11/8, 11/8) and perpendicularly a point(9/4, 1/2) on the curve. Using distance formula:

=> √(11/8 - 9/4)² + (11/8 - 1/2)²

=> 7√2/8