Question

How to find all the lines through the point (2,1) that are tangent to the parabola y=x^2?

Answer 1

if y = x^2, then dy/dx = 2x

So any tangent line on y = x^2 has a slope of 2x, assuming you're at the point (x,y), where y = x^2

Answer 2

This means that the slope is m = 2x

This line must go through (2,1), so that point helps us to uniquely define the line or lines

Answer 3

Any point on y = x^2 is of the form (x,y) which is really (x, x^2) since y = x^2

The tangent line must go through the two points

(x, x^2) and (2,1)

Answer 4

The tangent line has the slope m = 2x

so if we can connect the two points, and the slope, then we can solve for x

Answer 5

turns out we can since we can use the slope formula

m = (y2 - y1)/(x2 - x1)

m = (1 - x^2)/(2 - x)

2x = (1 - x^2)/(2 - x)

2x(2-x) = 1 - x^2

4x - 2x^2 = 1 - x^2

4x - 2x^2 - 1 + x^2 = 0

-x^2 + 4x - 1 = 0

x^2 - 4x + 1 = 0

Now use the quadratic formula to solve for x. I'll let you do this. Tell me what you get.