I will post a challenging math problem each day. The level of difficulty will vary, but most problems should not require any specialized knowledge beyond calculus. Problems are also posted on Twitter using the hashtag #mathpotd.
Tuesday, September 22, 2009
Double tangent line
Find the equation of the line that is tangent to the curve y = x4 − 14x3 + 69x2 at two points.
Let y = ax + b be the equation of the double tangent line. Then x^4 − 14x^3 + 69x^2 − ax − b has two double roots, so it can be factored as (x-p)^2 (x-q)^2.
Expanding and equating coefficients gives the following system of equations.
2p + 2q = 14
p^2 + 4pq + q^2 = 69
2pq^2 + 2p^2q = −a
p^2 q^2 = −b
Solving the first pair of equations yields (p,q) = (2,5) or (5,2). Substituting into the second pair of equations yields a = 140 and b = −100.
Found this a slightly different way. Since the slope m, of the line y=mx+b, is tangent to f(x)=x^4 − 14x^3 + 69x^2, m must be the same as the first derivative of f(x) at these intersections. Since f(x)=f'(x)x-b, we can infer that b = f(x)-f'(x)*x.
I then used a graphing calculator to graph the [m,b] values of the tangent line at each point along the graph: x(t) = [ f'(x), f(x)-f'(x)*x ]. This parametric line intersects itself at [140,-100], which gives the line y=140x-100.
The answer is y = 140x - 100
ReplyDeleteLet y = ax + b be the equation of the double tangent line. Then x^4 − 14x^3 + 69x^2 − ax − b has two double roots, so it can be factored as (x-p)^2 (x-q)^2.
Expanding and equating coefficients gives the following system of equations.
2p + 2q = 14
p^2 + 4pq + q^2 = 69
2pq^2 + 2p^2q = −a
p^2 q^2 = −b
Solving the first pair of equations yields (p,q) = (2,5) or (5,2). Substituting into the second pair of equations yields a = 140 and b = −100.
Found this a slightly different way. Since the slope m, of the line y=mx+b, is tangent to f(x)=x^4 − 14x^3 + 69x^2, m must be the same as the first derivative of f(x) at these intersections. Since f(x)=f'(x)x-b, we can infer that b = f(x)-f'(x)*x.
ReplyDeleteI then used a graphing calculator to graph the [m,b] values of the tangent line at each point along the graph: x(t) = [ f'(x), f(x)-f'(x)*x ]. This parametric line intersects itself at [140,-100], which gives the line y=140x-100.