If P and the origin are the points of intersection of the parabolas y2=32 x and 2 x2=27 y; and if θ is the acute angle between these curves at P, then 5 √ tan θ=

Question

If P and the origin are the points of intersection of the parabolas y2=32 x and 2 x2=27 y; and if θ is the acute angle between these curves at P, then 5 √ tan θ= (A) 2 (B) 2 √3 (C) 3 √2 (D) 3

Tardigrade Admin · Accepted Answer

Points of intersection of curves y2 =32 x ...(i) and 2 x2 =27 y ...(ii) From Eqs. (i), and (ii), we get 2 ⋅((y2/32))2 =27 y 2 ⋅ y4 =27 ⋅ 32 ⋅ 32 ⋅ y y =0, y3=512 ⋅ 27 y =24 From Eq. (i), we get x=0 x=18 So, coordinate of P(18,24). Equation of tangent through point P(18,24) on the curve y2=32 x ⇒ y ⋅ 24 =16(x+18) ⇒ 3 y =2 x+36 ∴ Slope m1 =2 / 3 Again equation of tangent through point P(18,24) on the 2 x2=27 y ⇒ 2 x ⋅ 18 =27 ((y+24)/2) ⇒ 8 x =3 y+72 ⇒ 3 y =8 x-72 Slope, m2=8 / 3 Angle between these curve θ= Angle between tangents drawn from points P tan θ =(m2-m1/1+m1 m2)=((8/3)-(2/3)/1+(8)3 ⋅ (2)3 tan θ =(6 / 3/(25)9)=(18/25) So, 5 √/ tan θ) =5 ⋅ √(18/25) =5 ⋅ (√18/5)=3 √2.

Q. If $P$ and the origin are the points of intersection of the parabolas $y^{2} = 32 x$ and $2 x^{2} = 27 y$ ; and if $θ$ is the acute angle between these curves at $P$ , then $5 tan θ =$

2

$23$

$32$

3

Q. If P and the origin are the points of intersection of the parabolas y2=32x and 2x2=27y; and if θ is the acute angle between these curves at P, then 5tanθ​=

2

23​

32​

3

Q. If $P$ and the origin are the points of intersection of the parabolas $y^{2} = 32 x$ and $2 x^{2} = 27 y$ ; and if $θ$ is the acute angle between these curves at $P$ , then $5 tan θ =$

$23$

$32$