Question:

Step 1: Rewrite the hyperbola in standard form. The given hyperbola is $x^2 - 4y^2 = 4$. Dividing both sides by 4: $$\frac{x^2}{4} - y^2 = 1$$ This is now in the standard form of a hyperbola. Step 2: Find the slope of the tangent using implicit differentiation. Differentiating the equation $x^2 - 4y^2 = 4$ implicitly with respect to $x$: $$2x - 8y\frac{dy}{dx} = 0$$ Solving for $\frac{dy}{dx}$: $$\frac{dy}{dx} = \frac{x}{4y}$$ This represents the slope of the tangent at any point on the hyperbola. Step 3: Determine the slope of the tangent from the normal's slope. We are given that the normal has slope 1. Since the normal and tangent are perpendicular, the slope of the tangent is the negative reciprocal of the normal's slope: $$\text{Slope of tangent} = -\frac{1}{1} = -1$$ Step 4: Find the point of contact on the hyperbola. Setting the tangent slope equal to $-1$: $$\frac{x}{4y} = -1$$ This gives us: $$x = -4y$$ Substituting this into the hyperbola equation $x^2 - 4y^2 = 4$: $$(-4y)^2 - 4y^2 = 4$$ $$16y^2 - 4y^2 = 4$$ $$12y^2 = 4$$ $$y^2 = \frac{1}{3}$$ $$y = \pm\frac{1}{\sqrt{3}}$$ Step 5: Calculate the length of the sub-tangent. The length of the sub-tangent is given by the formula: $$\text{Length of sub-tangent} = \left|\frac{y}{dy/dx}\right|$$ Substituting $\frac{dy}{dx} = \frac{x}{4y}$: $$\text{Length of sub-tangent} = \left|\frac{y}{\frac{x}{4y}}\right| = \left|\frac{4y^2}{x}\right|$$ Using $x = -4y$: $$\text{Length of sub-tangent} = \left|\frac{4y^2}{-4y}\right| = \left|\frac{4y^2}{-4y}\right| = |-y| = |y| = \frac{1}{\sqrt{3}}$$ Step 6: Find the value of $k$. We are given that the length of the sub-tangent equals $\frac{1}{\sqrt{k}}$: $$\frac{1}{\sqrt{k}} = \frac{1}{\sqrt{3}}$$ Therefore: $$k = 3$$ **Final Answer:** The value of $k$ is **3**, which corresponds to **Option 1: (c) 3**.