Question:

Step 1: Determine the slope of the tangent. The tangent makes an angle of $\pi/4$ with the x-axis. The slope $m$ is given by: $$m = \tan(\pi/4) = 1$$ Step 2: Write the equation of the tangent to the parabola. For the parabola $y^2 = 4ax$, the equation of a tangent with slope $m$ is: $$y = mx + \frac{a}{m}$$ Substituting $m=1$, the equation of the tangent becomes: $$y = x + a$$ This can be rewritten as $x - y + a = 0$. Step 3: Apply the tangency condition for the circle. The line $x - y + a = 0$ is tangent to the circle $x^2 + y^2 = c^2$. For a line to be tangent to a circle centered at the origin, the perpendicular distance from the origin $(0,0)$ to the line must be equal to the radius $c$. Step 4: Calculate the distance from the origin to the tangent line. The distance $d$ from the point $(x_0, y_0) = (0,0)$ to the line $Ax + By + C = 0$ is given by the formula: $$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$$ For the line $x - y + a = 0$, we have $A=1$, $B=-1$, $C=a$. $$d = \frac{|1(0) + (-1)(0) + a|}{\sqrt{1^2 + (-1)^2}} = \frac{|a|}{\sqrt{1+1}} = \frac{|a|}{\sqrt{2}}$$ Since $a > 0$ is given, the distance is: $$d = \frac{a}{\sqrt{2}}$$ Step 5: Equate the distance to the radius and determine the relationship between $a$ and $c$. For tangency, the distance $d$ must be equal to the radius $c$: $$\frac{a}{\sqrt{2}} = c$$ Multiplying both sides by $\sqrt{2}$: $$a = c\sqrt{2}$$