1
$\begingroup$

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?

$\endgroup$
8
  • 3
    $\begingroup$ Can you perhaps explain what you're trying to do with a line and an ellipse? $\endgroup$ – J. M.'s ennui Nov 17 '19 at 13:49
  • $\begingroup$ The line indicates that a focus is at (1,0). That does not seem correct. $\endgroup$ – Daniel Lichtblau Nov 17 '19 at 15:44
  • $\begingroup$ @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 $\endgroup$ – kile Nov 18 '19 at 2:56
  • $\begingroup$ Please include that description in your question. $\endgroup$ – J. M.'s ennui Nov 18 '19 at 3:01
  • $\begingroup$ Alright. @J.M.willbebacksoon $\endgroup$ – kile Nov 18 '19 at 3:08
5
$\begingroup$

enter image description here

M = {4, 3 k};
P = {1, 3/2};
{A, B} = Values@Solve[{x^2/4 + y^2/3 == 1, y == k (x - 1)}, {x, y}]
slope[A_, B_] := 1/Divide @@ (B - A);
{k1, k2, k3} = slope @@@ {{A, P}, {B, P}, {M, P}} // FullSimplify
FullSimplify[k1 + k2]

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.