The equation of the tangent to the parabola y2=4x whose slope is positive and which also touches x2+y2=(1/2) is

Question

The equation of the tangent to the parabola y2=4x whose slope is positive and which also touches x2+y2=(1/2) is (A) y=x+1 (B) y=2x+1 (C) x+y=2 (D) y=4x+(1/2)

Tardigrade Admin · Accepted Answer

Tangent to parabola is y=mx+(1/m) Tangent to circle is y=mx±(1/√2)√1 + m2 For common tangent (1/m2)=(1/2)(1 + m2)⇒ m4+m2-2=0 ⇒ m2=1⇒ m=1

Q. The equation of the tangent to the parabola $y^{2} = 4 x$ whose slope is positive and which also touches $x^{2} + y^{2} = \frac{1}{2}$ is

$y = x + 1$

$y = 2 x + 1$

$x + y = 2$

$y = 4 x + \frac{1}{2}$

Tangent to parabola is $y = m x + \frac{1}{m}$
Tangent to circle is $y = m x \pm \frac{1}{2} 1 + m^{2}$
For common tangent
$\frac{1}{m ^{2}} = \frac{1}{2} (1 + m^{2}) \Rightarrow m^{4} + m^{2} - 2 = 0$
$\Rightarrow m^{2} = 1 \Rightarrow m = 1$

Q. The equation of the tangent to the parabola y2=4x whose slope is positive and which also touches x2+y2=21​ is

y=x+1

y=2x+1

x+y=2

y=4x+21​