Consider the graph of the parabola given by the function f(x) = x^2 and let (a,
ID: 1720457 • Letter: C
Question
Consider the graph of the parabola given by the function f(x) = x^2 and let (a, b) be any point in the plane. The problem is to deride how many lines, if any, can be drawn from the point (a, b) so that the lines are tangent to the graph of the parabola. Draw the graph of f(x). Take some sample points P and try to draw a line from P that is tangent to the parabola at some point. Make a conjecture that describes when this can be done and when it cannot be done. Then refine your conjecture to guess how many tangent lines to the parabola can be drawn from a given point P. Let (r, r^2) be an arbitrary point on the parabola f(x) = x^2. Show that the equation of the tangent line to the graph of f(x) at (r, r^2) is given by y = 2rx - r^2. Let P = (a, b) be an arbitrary point in the plane. Show that a line can be drawn through P that is tangent to the parabola precisely when one can find a real number r such that (a, b) lies N on the line y = 2rx - r^2. Show that you must solve the quadratic equation r^2 - 2ar + b = 0.Explanation / Answer
Point P = (2,4)
slope of tangent = 2x =2r
Hence equation is y=2rx-r^2 (Use point slope formula)
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If (r,r^2) be any arbitrary point on the plane, then for the point to lie on the parabola
the line y = 2rx-r^2 must intersect y = x^2 at some point
i.e. eliminate y to have x^2 -2ax+b =0 should have real roots
This is possible when 4a^2-4b>0
i.e. when b <=a^2
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So when the point lies on the parabola there is one tangent i.e. when b = a^2
When the point lies outside i.e. when b <a^2 there are two tangents
When the point lies inside i.e. when b>a^2 no tangent.