Solving for Tangent Lines: Analytical and Graphical Approaches

[brochesspro]

Homework Statement

Find the equations of the two lines through the point $(3, 1)$ that are tangent to the curve $y = x^2 - 4$. Hint: Draw the graph, let $(a, a^2 - 4)$ be the point of tangency, and find $a$.

Relevant Equations

Given below.

I did it graphically by using GeoGebra.

My question is that what can I do to solve it analytically/algebraically. I used the point-slope formula and obtained $$\frac {y - (a^2-4)} {x - a} = 2a$$, which implies that $y = (2a)x + (-a^2-4)$.

I am not sure how to proceed from here onwards. Please help me solve this problem. I will see you in about 7 and a half hours.

 

[vela]

That gives you the equation of a line tangent to the parabola for $x=a$. (Check your work. You made a sign mistake.) Now you need to use the fact that you only want the lines that also pass through the point (3,1). That will allow you to determine which specific values of $a$ work.

 

[brochesspro]

That gives you the equation of a line tangent to the parabola at any point. (Check your work. You made a sign mistake.) Now you need to use the fact that you only want the lines that pass through the point (3,1). That will allow you to determine which specific values of $a$ work.

I see, I got the required points and the question is solved. Thank you.