Question

Equation of a line which is tangent to both the curves $y=x^2+1$ and $y=-x^2+1$ is

• A. $y=\sqrt{2}x+\frac{1}{2}$
• B. $y=\sqrt{2}x-\frac{1}{2}$
• C. $y=-\sqrt{2}x+\frac{1}{2}$
• D. $y=-\sqrt{2}x-\frac{1}{2}$

Answer 1

The correct option is A $y=\sqrt{2}x+\frac{1}{2}$

Given,

Equation of Parabolas

$P_1:y=x^2+1$

$P_2:y=-x^2+1$

$\text{Equation of tangent in}{P}_1$

$y-1=mx-\frac{1}{4}{m}^2$ - (i)

$\text{Equation of tangent in}{P}_2$

$(y-1)=mx+\frac{1}{4}{m}^2$ - (ii)

$\text{Equating equation (i) and (ii)}$

$mx-\frac{1}{4}{m}^2=mx+\frac{1}{4}{m}^2$

$\frac{1}{2}{m}^2=0$

$m=0$

y-1=0

y=1 is tangent common to both parabola.