Tangents And Normals

The equation of the curve Now, equation of the tangent at is

Step 1: Understanding the Concept:
A line is normal to a curve at a point if its slope is the negative reciprocal of the tangent's slope at that point. Additionally, the point of tangency must satisfy both the equation of the curve and the equation of the line.

Step 2: Key Formula or Approach:
1. Slope of normal ( ) Slope of tangent ( ) = .
2. Point (1, 3) must satisfy .

Step 3: Detailed Explanation:
1. Find slope of the normal: From , . The slope of the normal .
2. Find slope of the tangent: Since , the tangent slope .
3. Relate to curve derivative: . At , .
4. Use the point (1, 3) on the curve:
5. Calculate : .

Step 4: Final Answer
The value of is -15.