gaokao 2022 Q22

gaokao · China · national-A-arts 10 marks Parametric curves and Cartesian conversion
[Elective 4-4: Coordinate Systems and Parametric Equations] In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \frac { 2 + t } { 6 } \\ y = \sqrt { t } \end{array} \right.$ ($t$ is the parameter), and the parametric equation of curve $C _ { 2 }$ is $\left\{ \begin{array} { l } x = - \frac { 2 + s } { 6 } \\ y = - \sqrt { s } \end{array} \right.$ ($s$ is the parameter).
(1) Write the ordinary equation of $C _ { 1 }$ ;
(2) With the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 3 }$ is $2 \cos \theta - \sin \theta = 0$ . Find the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 1 }$, and the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 2 }$ .
[Elective 4-4: Coordinate Systems and Parametric Equations]\\
In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \frac { 2 + t } { 6 } \\ y = \sqrt { t } \end{array} \right.$ ($t$ is the parameter), and the parametric equation of curve $C _ { 2 }$ is $\left\{ \begin{array} { l } x = - \frac { 2 + s } { 6 } \\ y = - \sqrt { s } \end{array} \right.$ ($s$ is the parameter).\\
(1) Write the ordinary equation of $C _ { 1 }$ ;\\
(2) With the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 3 }$ is $2 \cos \theta - \sin \theta = 0$ . Find the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 1 }$, and the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 2 }$ .