Cho parabol \((P):{y^2} = 2x\), \((d):x - 2y + 6 = 0\). Khi đó: a) Đường chuẩn

a) Sai

b) Đúng

c) Sai

d) Đúng

 

a) Ta có \({y^2} = 2x\) có dạng: \({y^2} = 2px \Rightarrow 2p = 2 \Rightarrow \frac{p}{2} = \frac{1}{2} \Rightarrow F\left( {\frac{1}{2};0} \right) \Rightarrow x =  - \frac{1}{2}\).

b) Gọi \(M\left( {\frac{{{m^2}}}{2};m} \right) \in (P);d(M;d) = \frac{{\left| {\frac{{{m^2}}}{2} - 2m + 6} \right|}}{{\sqrt 5 }} = \frac{1}{{2\sqrt 5 }}\left[ {{m^2} - 4m + 12} \right]\)

\(\begin{array}{l} = \frac{1}{{2\sqrt 5 }}\left[ {{{(m - 2)}^2} + 8} \right] = \frac{1}{{2\sqrt 5 }}{(m - 2)^2} + \frac{4}{{\sqrt 5 }} \cdot \\d{(M;d)_{\min }} = \frac{4}{{\sqrt 5 }} = \frac{{4\sqrt 5 }}{5} \Rightarrow m = 2\end{array}\).