Tangents And Normals

Step 1:
The equation of the curve is given by:
We are asked to find the value of such that this curve touches the straight line at the point .
Step 2:
Substituting the point into the equation of the curve: This simplifies to: which holds true for any .
Step 3:
To ensure the curve touches the straight line at the point , the curve must be in a form that can represent a degenerate conic section (like an ellipse or hyperbola) that meets the straight line exactly at one point. This is achieved when , which corresponds to the equation of an ellipse, and the curve touches the line.
Step 4:
Therefore, the correct value of is .

The equation of the given circle represents two circles, one with and another with .
This equation can be rewritten as: Now, to find the equation of the circle that touches these two circles and has a radius of 3, we use the general formula for the equation of a circle: where is the center of the circle and is its radius. By solving this geometrically or using the condition for tangency, we find that the correct equation is or .