Overview
This lecture explains how to find equations for lines through the origin that are tangent to a given parabola, combining concepts of tangent lines and algebraic solutions.
Problem Setup
- The task is to find all lines passing through the origin that are tangent to a specific parabola.
- Tangent lines have the same formula as linear approximations: ( y = f'(a)(x - a) + f(a) ).
- The point of tangency is denoted as ( x = a ).
Equation for Tangent Line
- For a tangent at ( x = a ), the equation is ( y = f'(a)(x - a) + f(a) ).
- To ensure the line passes through the origin, substitute ( x = 0 ), ( y = 0 ) into the tangent line equation.
Solving for Points of Tangency
- Set ( 0 = f'(a)(0 - a) + f(a) ) to find possible values of ( a ).
- Substitute the parabola equation into ( f(a) ) and compute ( f'(a) ) (the derivative).
- For the parabola ( f(x) = x^2 - 2x + 4 ), the derivative is ( f'(x) = 2x - 2 ).
- Plug into the tangent condition: ( 0 = (2a - 2)(-a) + (a^2 - 2a + 4) ).
Algebraic Solution
- Expand and simplify: ( 0 = -2a^2 + 2a + a^2 - 2a + 4 ).
- Combine like terms: ( 0 = -a^2 + 4 ).
- Solve for ( a ): ( a^2 = 4 ), so ( a = 2 ) or ( a = -2 ).
- The x-values where tangent lines through the origin touch the parabola are ( x = 2 ) and ( x = -2 ).
Visual Confirmation
- Sketching confirms tangent lines at ( x = 2 ) and ( x = -2 ) pass through the origin.
- Other points on the parabola do not yield tangent lines passing through the origin.
Key Terms & Definitions
- Tangent Line — A line that touches a curve at exactly one point without crossing it.
- Parabola — A symmetric, U-shaped curve defined by a quadratic equation.
- Derivative ( f'(a) ) — The slope of the curve at a specific point ( x = a ).
- Point of Tangency — The point where a tangent line touches a curve.
Action Items / Next Steps
- Practice finding tangent lines through specific points for other curves.
- Review how to compute derivatives and apply the tangent line formula.
- Complete any assigned problems using these steps.