Question

If the x-intercept of a focal chord of the parabola y² = 8x + 4y + 4 is 3, then the length of this chord is equal to ______.

Answer 1

Given y²=8x+4y+4

⇒y²−4y=8x+4      

⇒y²−4y+4=8x+8

⇒(y−2)²=8(x+1)

Parabola with vertex V(-1, 2) and 4a=8
   ⇒ a = 2

So focus S(a+α,β)=S(2−1,2)=(1,2)

Let the equation of focal chord of the parabola x/a+y/b=1 and given x-intercept of focal chord is ‘3’. 

so x/3+y/b=1 and passing through focus (1, 2) 

so 1/3+2/b=1

2/b=1−1/3=2/3⇒b=3

Equation of focal chord x+y=3

So, slope of focal chord

∴ m = −1

tanθ = −1

⇒ θ= 3π/4
 
We know that length of focal chord in 4acosec²θ = 4(2)(√2)2 = 8(2) = 16

Answer 2

y² = 8x + 4y + 4

(y – 2)² = 8(x + 1)

y² = 4ax

a = 2, X = x + 1, Y = y – 2

focus (1, 2)

y – 2 = m (x – 1)

Put (3, 0) in the above line

m = – 1

Length of focal chord = 16