If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then |c| is equal to :

Question

If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then |c| is equal to : (A) 2 √3 (B) 8 √3 (C) 10 √3 (D) 16 √3

Tardigrade Admin · Accepted Answer

C =-29 m -9 m 3 a =2 Given ( at 2- a )2+4 a 2 t 2=64 arrow( a ( t 2+1))=8 ⇒ t 2+1=4= t 2=3 ⇒ t=√3 ∴ C=-a[-2 t-t3]=2 a t(2+t2) =2 √3(5) |C|=10 √3

Q. If $y = m x + c$ is the normal at a point on the parabola $y^{2} = 8 x$ whose focal distance is $8$ units, then $∣ c ∣$ is equal to :

$23$

$83$

$103$

$163$

$C = - 29 m - 9 m^{3}$
$a = 2$
Given $(a t^{2} - a)^{2} + 4 a^{2} t^{2} = 64$
$\to (a (t^{2} + 1)) = 8$
$\Rightarrow t^{2} + 1 = 4 = t^{2} = 3$
$\Rightarrow t = 3$
$∴ C = - a [- 2 t - t^{3}] = 2 a t (2 + t^{2})$
$= 23 (5)$
$∣ C ∣ = 103$

Q. If y=mx+c is the normal at a point on the parabola y2=8x whose focal distance is 8 units, then ∣c∣ is equal to :

23​

83​

103​

163​