The equation of the latus rectum of a parabola is given by y = 3.

Question

The axis of the parabola is x = 0, and its vertex is (0,0).

Find the length of the focal chord that meets the parabola (2,-1/3)

Answer 1

Latus rectum is the line segment through a focus of a conic section, perpendicular to the major axis which has both end points on the curve.

Latus rectum y = 3 is a horizontal line above x axis.So the parabola opens upward. (2, -1/3) not a point on the parabola.

If y = 3 is a directrix, then it has solution.

 

If the directrix y = 3

The parabola is opens downward.

It's equation is (x - h)² = 4p(y - k)

Where center = (h, k).

Focus (h, k + p)

And it's directrix equation is y = k - p

In this case (h, k) = (0, 0)

y = 3

k - p = 3

Substitute the k value.

0 - p = 3

p = - 3

So focus (h, k + p) = (0 , - 3)

Now the parabola equation is (x - 0)² = 4(-3)(y - 0)

x² = - 12y ---> (1)

Let the points are (x₁, y₁) = (0, -3) and (x₂, y₂) = (2, -1/3)

The line equation y - y₁ = [(y₂ - y₁)/(x₂ - x₁)] (x - x₁)

y + 3 = [(-1/3) + 3)/(2 - 0)] (x - 0)

y + 3 = [(-1/3) + 3)/(2 - 0)] (x - 0)

y + 3 = (8/6)x

y = (4/3)x - 3 ---> (2)

Now find the point of intersection of the line and parabola.

Substitute y = (4/3)x - 3 in (1).

x² = - 12[(4/3)x - 3]

 

x² = - 16x + 36

x² + 16x - 36 = 0

x² + 18x - 2x - 36 = 0

x(x + 18) - 2(x  + 18) = 0

(x + 18)(x - 2) = 0

x = - 18, x = 2

Substitute x values in (2).

y = (4/3)(- 18) - 3

y = - 24 - 3 = - 27

y = (4/3)( 2) - 3

y = (8/3) - 3 = -1/3

 

To find the length of the focal chord that meets the parabola. just calculate the distance between the points.

(x₁ , y₁) = (- 18, - 27) and (x₂, y₂) = (2, -1/3)

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √{(2 + 18)² + [(-1/3) - (- 27)]²}

= √{(20)² + [(-1/3) + 27)²]}

= √[400 + (80/3)²]

= √[400 + (6400/9)]

= √(3600 + 6400)/9

= √(10000/9)

= 100/3

Length of the focal chord = 33.33.