Tangents And Normals

Concept: Normal equation + geometric condition.

Step 1: Let curve be and slope of tangent = .
Slope of normal:

Step 2: Equation of normal at .


Step 3: Find intercept on -axis.
Put :

Step 4: Use given condition.
Square of ordinate = twice rectangle:

Step 5: Form differential equation.


Step 6: Solve differential equation.
Rearranging and integrating leads to:

Step 7: Use given point .


Step 8: Final equation.
Conclusion:
Answer = Option (D)

Concept: Implicit differentiation to find slope of tangent.

Step 1: Differentiate implicitly.


Step 2: Find slope at .


Step 3: Equation of tangent.


Step 4: Simplify.


Step 5: Rearrange.
Conclusion:
Answer = Option (C)

Concept: Equation of tangent + intercept form → area of triangle.

Step 1: Differentiate function.


Step 2: Slope at .


Step 3: Equation of tangent.


Step 4: Find intercepts.
-intercept: -intercept:

Step 5: Area of triangle.


Step 6: Solve equation.
Simplifying gives: Conclusion: