Question

The number of common tangents that can be drawn to the curves $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $x^2+y^2=16$ is

• A. 0
• B. 1
• C. 3
• D. 2

Hint:

To find common tangents, it's often helpful to sketch the curves and analyze their relative positions. If the curves touch at a point, they share a common tangent at that point. The number of common tangents depends on whether the curves intersect, touch, or are separate.

Answer 1

The Correct Option is D

Step 1: Identify the conic sections and their properties.
The first curve is a hyperbola: $\frac{x^2}{16}-\frac{y^2}{9}=1$. Its parameters are $a^2=16, b^2=9$. The vertices are at $(\pm a, 0) = (\pm 4, 0)$. The second curve is a circle: $x^2+y^2=16$. Its center is at the origin $(0,0)$ and its radius is $R = \sqrt{16} = 4$.

Step 2: Analyze the relative positions of the circle and the hyperbola.
The vertices of the hyperbola are the points $(\pm 4, 0)$. The x-intercepts of the circle are found by setting $y=0$, which gives $x^2=16 \implies x = \pm 4$. Since the vertices of the hyperbola coincide with the x-intercepts of the circle, the two curves touch at these two points: $(4,0)$ and $(-4,0)$.

Step 3: Find the tangent at a point of contact.
Let's find the slope of the tangent to each curve at the point of contact $(4,0)$. For the circle $x^2+y^2=16$, implicit differentiation gives $2x + 2y \frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$. At $(4,0)$, the slope is undefined, which means the tangent line is vertical. The equation of the tangent is $x=4$. For the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$, implicit differentiation gives $\frac{2x}{16} - \frac{2y}{9}\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = \frac{9x}{8y}$. At $(4,0)$, the slope is also undefined, indicating a vertical tangent line, $x=4$. Since both curves have the same tangent line, $x=4$, at their point of contact $(4,0)$, this line is a common tangent.

Step 4: Consider the other point of contact.
By symmetry, the same analysis applies to the point $(-4,0)$. The tangent to both the circle and the hyperbola at $(-4,0)$ is the vertical line $x=-4$. This is the second common tangent.

Step 5: Conclude the total number of common tangents.
The circle lies between the two branches of the hyperbola, touching each at its vertex. Any other tangent to the circle would have to cross the hyperbola. Therefore, there are no other common tangents. The two common tangents are $x=4$ and $x=-4$. The total number of common tangents is 2.