#### To determine

**To show:**

The sum of x- coordinate of normal lines of parabola intersecting at a common point is 0

#### Answer

The sum of x- coordinate of normal lines of parabola intersecting at a common point is 0

#### Explanation

**1) Formula:**

i. Power rule:

ddxxn=nxn-1

ii. Tangent normal formula: If m1 is the slope of tangent and m2 is the slope of normal then,

m1*m2=-1

iii. Point-slope formula:

y-y1=m(x-x1)

**2) Calculations:**

Equation of parabola is, y=x2

Consider the point (a, a2) satisfies the equation of parabola

Now we find the slope of parabola,

y=x2

Differentiate with respect to x,

dydx=ddx(x2)

By using power rule,

dydx=2x

At point (a, a2) slope of tangent is,

dydx=m1=2a

Let m2 be slope of tangent line at (a, a2). Now by using relation between slope of tangent and normal line,

2a*m2=-1

⇒m2=-12a

Therefore, by using point-slope formula the equation of normal line is

y-a2= -12a(x-a)

2ay-2a3= -(x-a)

⇒2ay-2a3+x-a=0

⇒2a3-2ay-x+a=0

Divide by 2,

⇒a3-ay-x2+a2=0

Let (x, y) be the point of intersection of three normal lines. Then we can obtain the x-coordinates a of the points on the curve through which each of the line passes by solving the cubic equation. . Thus it follows that the x-coordinates of the three points are

Therefore, sum of roots is equal to coefficient of a2

Here coefficient of a2 is 0

Therefore,

If a1, a2 and a3 are roots then a1+a2+a3=0 But roots are x coordinate of the normal line

Therefore,

The sum of x- coordinate of normal lines of parabola intersecting at a common point is 0

**Conclusion:**

The sum of x- coordinate of normal lines of parabola intersecting at a common point is 0