HMMT 二月 1998 · CALC 赛 · 第 8 题
HMMT February 1998 — CALC Round — Problem 8
题目详情
英文原题
Question Eight . [6 points]
Find the slopes of all lines passing through the origin and tangent to
2 3
the curve y = x + 39 x − 35.
解析
英文解析
- Problem: Find the slopes of all lines passing through the origin and tangent to the curve
2 3
y = x + 39 x − 35.
Solution: Any line passing throug the origin has equation y = mx , where m is the slope of the line. If adyline is tangent to the given curve, then at the point of tangency, ( x, y ), = m .
2 dxdy dy
2 3 x +39
First, we calculate of the curve: 2 ydy = 3 x dx + 39 dx ⇒ = . Substituting mx for y , we getdx dx 2 ythe following system of equations:
2 2 3
m x = x + 39 x − 35
3 x + 392
m =
2 mx
Solving for x yields the equation x − 39 x + 70 = 0 ⇒ ( x − 2)( x + 7)( x − 5) = 0 ⇒ x = 2 or x = − 7 or 3
x = 5. These solutions indicate the x -coordinate of the points at which the desired lines are tangent to the
√
curve. Solving for the slopes of these lines, we get m = ± for x = 2, no real solutions for x = − 7, and 51
√ √ √2
285 51 285
m = ± for x = 5. Thus m = ± , ± .
5 2 5
∞
∑ 1