Question

The point on the curvey= x2 + 4x +3 which is closest to the line y = 3x + 2 ​

Answer 1

Answer:

(-1/2 , 5/4)

Step-by-step explanation:

Minimum distance is given when both are parallel to each other(even for a very short distance).

     In such cases,  both have same slopes

⇒ slope of curve = slope of line

⇒ tangent of curve = slope of line   ...(1)

On comparing y = 3x + 2 with y = mx + c, we get m = slope = 3

   Given curve is y = x² + 4x + 3

⇒ tangent = y'

                 = d(x² + 4x + 3)/dx

⇒ tangent = 2x + 4

Using (1), we get,     2x + 4 = 3

                                   x = -1/2

[x is the required x-coordinate of curve]

Substituting x in eq. of curve,   we get

       y = (-1/2)² + 4(-1/2) + 3

       y = 5/4

Hence the required point is (-1/2 , 5/4)

Answer 2

Given Data :

curve (y) = x² + 4x + 3

closest to the line y = 3x + 2

to find :

• The point which is closest to the curve

Solution :

✿ When The distance between two lines is Minimum that indicates, they are parallel to each other .

In that case, both have same Slopes

➻ slope of line = slope of curve

➻ Tangent of Curve = Slope of Line ____eq(1)

On comparing y = 3x + 2 with y = mx + c ,

we get

➛ m = slope of line = 3

As given that, curve = x² + 4x + 3

↬ Tangent = y'

↬ Tangent = d(x² + 4x + 3)/dx

↬ Tangent = 2x + 4

From Eq(1),

⇢ 2x + 4 = 3

⇢ x = -½

• [x is the required x coordinate]

finding y :

⇝ y = (-½)² + 4(-½) + 3

⇝ y = ¼ - ½ + 3

⇝ y = 5/4

hence, The Required point is (-½ , 5/4)