The line [tex]l[/tex] has equation
[tex]3 x-2 y=k[/tex]
where [tex]k[/tex] is a real constant.
Given that the line [tex]l[/tex] intersects the curve with equation
[tex]y=2 x^2-5[/tex]
at two distinct points, find the range of possible values for [tex]k[/tex].
Answer (2)
Express y in terms of x and k from the line equation: y = 2 3 x − k .
Substitute into the curve equation and rearrange to get a quadratic equation: 4 x 2 − 3 x + ( k − 10 ) = 0 .
Calculate the discriminant: 169 − 16 k .
Set the discriminant greater than zero and solve for k : k < 16 169 .
k < 16 169
Explanation
Problem Analysis We are given a line l with the equation 3 x − 2 y = k and a curve with the equation y = 2 x 2 − 5 . We need to find the range of possible values for k such that the line intersects the curve at two distinct points. This means we need to find the values of k for which the system of equations has two distinct solutions.
Express y in terms of x and k First, we express y in terms of x and k from the equation of the line: 3 x − 2 y = k ⇒ 2 y = 3 x − k ⇒ y = 2 3 x − k .
Substitute into the curve equation Next, we substitute this expression for y into the equation of the curve: 2 3 x − k = 2 x 2 − 5 . Multiplying both sides by 2, we get: 3 x − k = 4 x 2 − 10 . Rearranging the terms, we obtain a quadratic equation in x : 4 x 2 − 3 x + ( k − 10 ) = 0 .
Calculate the discriminant For the quadratic equation a x 2 + b x + c = 0 to have two distinct real roots, the discriminant b 2 − 4 a c must be greater than zero. In our case, a = 4 , b = − 3 , and c = k − 10 . Thus, the discriminant is: ( − 3 ) 2 − 4 ( 4 ) ( k − 10 ) = 9 − 16 ( k − 10 ) = 9 − 16 k + 160 = 169 − 16 k .
Set the discriminant greater than zero We require the discriminant to be greater than zero for two distinct intersection points: 0"> 169 − 16 k > 0 . Solving for k , we get: 16k \Rightarrow k < \frac{169}{16}"> 169 > 16 k ⇒ k < 16 169 .
Express the range of possible values for k Therefore, the range of possible values for k is k < 16 169 .
Final Answer The range of possible values for k is k < 16 169 .
Examples
Understanding the intersection of lines and curves is crucial in various fields. For instance, in physics, analyzing the trajectory of a projectile involves understanding how its parabolic path (a curve) intersects with a line representing a barrier or a target. Similarly, in economics, determining the equilibrium point in a market involves finding the intersection of supply and demand curves. In computer graphics, ray tracing algorithms rely on finding the intersection points of rays (lines) with various surfaces (curves) to render realistic images. Therefore, the ability to analyze the intersection of lines and curves is a fundamental skill with wide-ranging applications.
The line 3 x − 2 y = k intersects the curve y = 2 x 2 − 5 at two distinct points for values of k that are less than 16 169 . Therefore, the range is k < 16 169 .
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