giả sử \(\left(P\right):y=ax^2+bx+c\)
khi đó \(\left(P\right)\cap\left(d\right)\) \(\Leftrightarrow ax^2+bx+c=\left(3-2m\right)x-m^2-2m\)
\(\Leftrightarrow ax^2+\left(b+2m-3\right)x+m^2+2m+c=0\)
để \(\left(P\right)\) tiếp xúc \(\left(d\right)\) \(\Leftrightarrow\left(b+2m-3\right)^2-4a\left(m^2+2m+c\right)=0\)
\(\Leftrightarrow b^2+4m^2+9+4mb-12m-6b-4am^2-8am-4ac=0\)
\(\Leftrightarrow\left(b^2+9-6b-4ac\right)+\left(4b-12-8a\right)m+\left(4-4a\right)m^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}b^2+9-6b-4ac=0\\4b-12-8a=0\\4-4a=0\end{matrix}\right.\) \(\left\{{}\begin{matrix}a=1\\b^2-6b+9-4c\\4b-20=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=1\\b=5\\c=1\end{matrix}\right.\)
vậy tồn tại \(\left(P\right)y=x^2+5x+1\) \(\Rightarrow\left(đpcm\right)\)
