Respuesta :
Answer:
k = 11.
Step-by-step explanation:
y = x^2 - 5x + k
dy/dx = 2x - 5 = the slope of the tangent to the curve
The slope of the normal = -1/(2x - 5)
The line 3y + x =25 is normal to the curve so finding its slope:
3y = 25 - x
y = -1/3 x + 25/3 <------- Slope is -1/3
So at the point of intersection with the curve, if the line is normal to the curve:
-1/3 = -1 / (2x - 5)
2x - 5 = 3 giving x = 4.
Substituting for x in y = x^2 - 5x + k:
When x = 4, y = (4)^2 - 5*4 + k
y = 16 - 20 + k
so y = k - 4.
From the equation y = -1/3 x + 25/3, at x = 4
y = (-1/3)*4 + 25/3 = 21/3 = 7.
So y = k - 4 = 7
k = 7 + 4 = 11.
The value of k for which the line 3y+x=25 is a normal to the curve y=x^2-5x+k is 11
The equations of the line and the curve are given as:
[tex]3y + x = 25[/tex]
[tex]y = x^2 - 5x + k[/tex]
Differentiate the equation of the curve
[tex]y' =2x - 5[/tex]
The above equation represents the slope of the tangent to the curve
For a line normal to the curve, the slope is represented as:
[tex]m = -\frac 1{y'}[/tex]
So, we have:
[tex]m= -\frac{1}{2x - 5}[/tex]
Recall that:
[tex]3y + x = 25[/tex]
Make y the subject
[tex]y = -\frac 13x +\frac{25}3[/tex]
The slope of the above equation is:
[tex]m = -\frac 13[/tex]
Equate both slopes
[tex]-\frac{1}{3} = -\frac{1}{2x - 5}[/tex]
Multiply both sides by -1
[tex]\frac{1}{3} = \frac{1}{2x - 5}[/tex]
Cross multiply
[tex]2x - 5 = 3[/tex]
Add 5 to both sides
[tex]2x = 8[/tex]
Divide both sides by 2
[tex]x = 4[/tex]
Substitute 4 for x in [tex]y = x^2 - 5x + k[/tex] and [tex]y = -\frac 13x +\frac{25}3[/tex]
[tex]y = x^2 - 5x + k[/tex]
[tex]y =4^2 - 5(4) + k[/tex]
This gives
[tex]y =16 - 20 + k[/tex]
[tex]y =- 4 + k[/tex]
[tex]y = -\frac 13x +\frac{25}3[/tex]
[tex]y = -\frac 13 \times 4 +\frac{25}{3}[/tex]
[tex]y = -\frac 43 +\frac{25}{3}[/tex]
Take LCM
[tex]y =\frac{-4+25}{3}[/tex]
[tex]y =\frac{21}{3}[/tex]
[tex]y = 7[/tex]
Substitute 7 for y in [tex]y =- 4 + k[/tex]
[tex]7 = -4 + k[/tex]
Add 4 to both sides
[tex]11 = k[/tex]
Rewrite the equation as:
[tex]k = 11[/tex]
Hence, the value of k is 11
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