Let the left and right vertexes of hyperbola X & # 178; / 2-y & # 178; = 1 be a &;, a &;, P (X &;, Y &;), q (X &;, - Y &;) respectively, which are two different moving points on the hyperbola, and the equation for finding the locus e of the intersection of a &; P and a &; Q

Let the left and right vertexes of hyperbola X & # 178; / 2-y & # 178; = 1 be a &;, a &;, P (X &;, Y &;), q (X &;, - Y &;) respectively, which are two different moving points on the hyperbola, and the equation for finding the locus e of the intersection of a &; P and a &; Q

P (X &;, Y &;) on hyperbola
x1²/2-y1²=1
y1²=(x1²-2）/2
A1（-√2,0）,A2（√2,0）
A1P； k=y1/(x1+√2)
Equation y = Y1 / (x1 + √ 2) * (x + √ 2) (1)
In the same way
The equation of a2q y = - Y1 / (x1 - √ 2) * (x - √ 2) (2)
（1）*（2）
y²=-y1²/(x1²-2)*(x²-2）=-1/2 (x²-2）
That is, X & # 178; - 2 = - 2Y & # 178;
That is, the equation of the locus e of the intersection X & # 178; + 2Y & # 178; = 2