Given ellipse C x 2 a 2 y 2 b 2 1 ( a b 0 ) Ellipse C passes through the point P (1,3 2) , and has eccentricity e 1 2 Given line x 4 AB is the chord that intersects line on point M , and F is the right focus of ellipse C The slopes of PA , P B , and PM are respectively k1 , k2 , and k3 Is there a constant that satisfies k1 k2 k3 If there is, please calculate P A'

Question: Given ellipse C : x 2 /a 2 +y 2 /b 2 =1 ( a>b>0 ). Ellipse C passes through the point P : (1,3/2)

Given ellipse $C$ : $x 2 /a 2 +y 2 /b 2 =1$ ( $a>b>0$ ). Ellipse $C$ passes through the point $P$ : $(1,3/2)$ , and has eccentricity $e=1/2$ . Given line $ℓ$ : $x=4$ . $AB$ is the chord that intersects line $ℓ$ on point $M$ , and $F$ is the right focus of ellipse $C$ . The slopes of $PA$ , P $B$ , and $PM$ are respectively $k1$ , $k2$ , and $k3$ . Is there a constant $λ$ that satisfies $k1+k2=λk3$ ? If there is, please calculate $λ$ .

P. A'

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