Ellipse and vectors
Given that ellipse has the equation: , and line always pass through point . intersects with the y-axis at point and the ellipse at point .
If , then will always be equal to . Find the value of .
The answer is -2020.
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1 solution
Choose line l to be a horizontal line through Q ( 2 0 , 0 ) . Then its equation is y = 0 , so that it intersects the y -axis at P ( 0 , 0 ) the ellipse at M ( − 2 0 2 0 , 0 ) and N ( − 2 0 2 0 , 0 ) .
That makes P M = ( − 2 0 2 0 , 0 ) , M Q = ( 2 0 2 0 + 2 0 , 0 ) , P N = ( 2 0 2 0 , 0 ) , and N Q = ( − 2 0 2 0 + 2 0 , 0 ) , so that λ 1 = M Q P M = 2 0 2 0 + 2 0 − 2 0 2 0 and λ 2 = N Q P N = − 2 0 2 0 + 2 0 2 0 2 0 , which means λ = λ 1 + λ 2 = 2 0 2 0 + 2 0 − 2 0 2 0 + − 2 0 2 0 + 2 0 2 0 2 0 = 1 0 0 0 2 0 2 0 .
Therefore, 1 0 0 0 λ = 1 0 0 0 ⋅ 1 0 0 0 2 0 2 0 = 2 0 2 0 .