If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :

Question

If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to : (A) -64 (B) 128 (C) -128 (D) -32

Tardigrade Admin · Accepted Answer

y = mx + 4 is tangent to y2 = 4x ⇒ m = (1/4) y = (1/4) x+4 is tangent to x2 = 2by ⇒ x2 - (b/2)x-8b = 0 ⇒ D = 0 b2 + 128b = 0 ⇒ b = - 128, 0 b≠ 0 ⇒ b = -128

Q. If $y = m x + 4$ is a tangent to both the parabolas, $y^{2} = 4 x$ and $x^{2} = 2 b y$ , then b is equal to :

-64

128

-128

-32

$y = m x + 4$ is tangent to $y^{2} = 4 x$
$\Rightarrow m = \frac{1}{4}$
$y = \frac{1}{4} x + 4$ is tangent to $x^{2} = 2 b y$
$\Rightarrow x^{2} - \frac{b}{2} x - 8 b = 0$
$\Rightarrow D = 0$
$b^{2} + 128 b = 0$
$\Rightarrow b = - 128, 0$
$b \neq = 0 \Rightarrow b = - 128$

Q. If y=mx+4 is a tangent to both the parabolas, y2=4x and x2=2by, then b is equal to :

-64

128

-128

-32

y=mx+4 is tangent to y2=4x ⇒m=41​ y=41​x+4 is tangent to x2=2by ⇒x2−2b​x−8b=0 ⇒D=0 b2+128b=0 ⇒b=−128,0 b=0⇒b=−128

Q. If $y = m x + 4$ is a tangent to both the parabolas, $y^{2} = 4 x$ and $x^{2} = 2 b y$ , then b is equal to :

$y = m x + 4$ is tangent to $y^{2} = 4 x$
$\Rightarrow m = \frac{1}{4}$
$y = \frac{1}{4} x + 4$ is tangent to $x^{2} = 2 b y$
$\Rightarrow x^{2} - \frac{b}{2} x - 8 b = 0$
$\Rightarrow D = 0$
$b^{2} + 128 b = 0$
$\Rightarrow b = - 128, 0$
$b \neq = 0 \Rightarrow b = - 128$