How can I simplify my geometric calculation?

Here is a math problem I am dealing with right now:

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 $$\ell$$: $$x=4$$. $$\overline{AB}$$ is the chord that intersects line $$\ell$$ on point $$M$$, and $$F$$ is the right focus of ellipse $$C$$. The slopes of $$\overline{PA}$$, $$\overline{PB}$$, and $$\overline{PM}$$ are respectively $$k_1$$, $$k_2$$, and $$k_3$$. Is there a constant $$\lambda$$ that satisfies $$k_1 + k_2 =\lambda k_3$$? If there is, please calculate $$\lambda$$.

How can use the last results into next input?

Solve[{x^2/4 + y^2/3 == 1, y == k (x - 1)}, {x, y}]

{x -> (2 (2 k^2 - 3 Sqrt[1 + k^2]))/(3 + 4 k^2),
y -> -k + (4 k^3)/(3 + 4 k^2) - (6 k Sqrt[1 + k^2])/(3 + 4 k^2)},
{x -> (2 (2 k^2 + 3 Sqrt[1 + k^2]))/(3 + 4 k^2),
y -> -k + (4 k^3)/(3 + 4 k^2) + (6 k Sqrt[1 + k^2])/(3 + 4 k^2)}}

A =
{(2 (2 k^2 - 3 Sqrt[1 + k^2]))/(3 + 4 k^2),
-k + (4 k^3)/(3 + 4 k^2) - (6 k Sqrt[1 + k^2])/(3 + 4 k^2)};
B =
{(2 (2 k^2 + 3 Sqrt[1 + k^2]))/(3 + 4 k^2),
-k + (4 k^3)/(3 + 4 k^2) + (6 k Sqrt[1 + k^2])/(3 + 4 k^2)};
M = {4, 3 k};
P = {1, 3/2};
Simplify[k1 = (A[[2]] - P[[2]])/(A[[1]] - P[[1]]), k ∈ Reals]
Simplify[k2 = (B[[2]] - P[[2]])/(B[[1]] - P[[1]])]
Simplify[k3 = (M[[2]] - P[[2]])/(M[[1]] - P[[1]])]
Simplify[k1 + k2]

(3 + 2 k + 4 k^2 + 4 k Sqrt[1 + k^2])/(2 + 4 Sqrt[1 + k^2])

(3 + 2 k + 4 k^2 - 4 k Sqrt[1 + k^2])/(2 - 4 Sqrt[1 + k^2])

-(1/2) + k

-1 + 2 k


How can I use the result of the last and put it in the next input?

• Can you perhaps explain what you're trying to do with a line and an ellipse? Nov 17, 2019 at 13:49
• The line indicates that a focus is at (1,0). That does not seem correct. Nov 17, 2019 at 15:44
• @J.M.willbebacksoon. It's a math problem I am dealing with right now. Ellipse C: x 2 /a 2 + y 2 /b 2 =1 (a>b>0). Ellipse C pass through point P (1,3/2), Eccentricity e=1/2. Line l: x=4. AB is the chord that intersect with line l on point M, And F is the right foci of ellipse C. The slopes of PA ,PB, PM are respectively k 1 , k 2 , k 3 . Is there a constant λ that meets k 1 + k 2 =λ k 3 ? If there is , please calculate λ
– kile
Nov 18, 2019 at 2:56
• Please include that description in your question. Nov 18, 2019 at 3:01
• Alright. @J.M.willbebacksoon
– kile
Nov 18, 2019 at 3:08

M = {4, 3 k};