If OAB is an equilateral triangle inscribed in the parabola y2=4ax with O as the vertex, then the length of the side of the Δ OAB is

Question

If OAB is an equilateral triangle inscribed in the parabola y2=4ax with O as the vertex, then the length of the side of the Δ OAB is (A)  8 a √3  (B)  4 a √3  (C)  2 a √3  (D)  a √3

Tardigrade Admin · Accepted Answer

In Δ OCA, tan 30o=(AC/OC) ⇒ (1/√3)=(2at/at2) ⇒ t=2√3 Again in Δ OCA, OA=√OC2+AC2 =√(at2)2+(2at)2 =√[(2√3)2]2a2+4a2(2√3)2 =√144 a2+48a2=√192 a2 =8√3 a

Q. If OAB is an equilateral triangle inscribed in the parabola $y^{2} = 4 a x$ with O as the vertex, then the length of the side of the $Δ O A B$ is

$8 a 3$

$4 a 3$

$2 a 3$

$a 3$

In $Δ OC A, tan 30^{o} = \frac{A C}{OC}$
$\Rightarrow$ $\frac{1}{3} = \frac{2 a t}{a t ^{2}}$
$\Rightarrow$ $t = 23$
Again in $Δ OC A,$
$O A = O C^{2} + A C^{2}$
$= (a t^{2})^{2} + (2 a t)^{2}$
$= [(23)^{2}]^{2} a^{2} + 4 a^{2} (23)^{2}$
$= 144 a^{2} + 48 a^{2} = 192 a^{2}$
$= 83 a$

Q. If OAB is an equilateral triangle inscribed in the parabola y2=4ax with O as the vertex, then the length of the side of the ΔOAB is

8a3​

4a3​

2a3​

a3​