gaokao 2023 Q5

gaokao · China · national-II 5 marks Conic sections Chord Properties and Midpoint Problems

Given the ellipse $\frac{x^2}{3}+y^2=1$ with left and right foci $F_1, F_2$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. If the area of $\triangle F_1AB$ is 2 times the area of $\triangle F_2AB$, then $m=$
A. $\frac{2}{3}$
B. $\frac{\sqrt{2}}{3}$
C. $-\frac{\sqrt{2}}{3}$
D. $-\frac{2}{3}$

C
According to the problem, $s_{\triangle F_1AB}=2s_{\triangle F_2AB}$. Let the distances from the left and right foci $F_1, F_2$ of the ellipse $\frac{x^2}{3}+y^2=1$ to the line $y=x+m$ be $d_1$ and $d_2$ respectively, with $-2

Given the ellipse $\frac{x^2}{3}+y^2=1$ with left and right foci $F_1, F_2$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. If the area of $\triangle F_1AB$ is 2 times the area of $\triangle F_2AB$, then $m=$\\
A. $\frac{2}{3}$\\
B. $\frac{\sqrt{2}}{3}$\\
C. $-\frac{\sqrt{2}}{3}$\\
D. $-\frac{2}{3}$

Paper Questions

Q1 Q2 Q5 Q6 Q7 Q8 Q9 Q10 Q11