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Sagot :
Answer:
h = 4k + 1 or -4k + 1
Step-by-step explanation:
When a line/curve is tangent to another line/curve, the discriminant (Δ) of their combined function will equal to 0. To find the h in terms of k, we can use this way:
First, we combine both equations:
[tex]\left\{\begin{aligned}y&=0\\y&=2x^2+(h-1)x+2k^2\end{aligned}[/tex]
Since y = y, then:
[tex]2x^2+(h-1)x+2k^2=0[/tex]
Next, we find the discriminant of the equation by using this formula:
[tex]\boxed{\Delta=b^2-4ac}[/tex]
Given:
- [tex]a=\bf 2[/tex]
- [tex]b= \bf (h - 1)[/tex]
- [tex]c = \bf 2k^2[/tex]
Hence:
[tex]\Delta=0\ \Longleftrightarrow \text{the line is tangent to the curve}[/tex]
[tex]b^2-4ac=0[/tex]
[tex](h-1)^2-4(2)(2k^2)=0[/tex]
[tex](h-1)^2=16k^2[/tex]
[tex]h-1=\sqrt{16k^2}[/tex]
[tex]h-1=\pm4k[/tex]
[tex]h=\pm4k+1[/tex]
Therefore:
[tex]\bf h=4k+1\ or\ -4k+1[/tex]
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