Question

find the equation of the normal of the parabola Y is equal to x square - 7 x + 8 which make the angle 45 degree with x axis

Answer 1

Parabola ( function ) is ,,,

y = x² - 7x + 8........(A)

diffrentiate w.r.t 'x',

dy/dx = 2x - 7

let Normal is at point P( x₁ , y₁ )

Now slope of normal at p, m = - 1 / [ dy/dx] at P

=> m= -1/ (2x₁ - 7 ) ..........〈1〉

also, slope of normal, m = tan45⁰

m = 1 ..............〈2〉

from (1) and (2)

-1/ ( 2x₁ - 7 ) = 1

2x ₁ - 7 = -1
2x₁ = 7-1
2x ₁=6
x₁= 3

for x ₁ = 3, y₁= 3*3 - 7 *3 + 8 = -4
point P (3, -4)
slope=1

equation of normal,

(y - y₁ ) = m ( x - x₁ )
y +4 = 1 ( x -3 )

x - y - 7 =0