So the equation is x? =4y
2.
Let the p coordinate be (m, m? /4), and the q coordinate is (n, n? /4)
∵PF⊥QF has (1-m? /4)/-m * ( 1-n？ /4)/-n =- 1①
The slope of the tangent PQ of PQ and parabola at point p ⊥. is (n? /4-m？ /4)/(n-m) is perpendicular to the tangent at p, which means that the tangent slope at p is -(n-m)/(n? /4-m？ /4)
Substitution gives the tangent equation at p as y-m /4=-(n-m)(x-m)/(n? /4-m？ /4)=-4(x-m)(n-m)/(n-m)(n+m)=-4(x-m)/(m+n)②
( 1-m/2)( 1+m/2)n/( 1-n/2)( 1+n/2)m =- 1。
n-nm？ /4=-m+mn？ /4
(n+m)-nm(n+m)/4=0
(n+m)( 1-mn/4)=0
Get n=-m or mn=4④.
Then it needs to be discussed in two situations. ② The straight line is tangent to the parabola, and ② it is substituted into the parabolic equation.
y=x？ /4=m？ /4-4(x-m)/(m+n)
x？ /4+4(x-m)/(m+n)-m？ /4=0
x？ /4+4x/(m+n)-4m/(m+n)-m？ /4=0
Do tangent need to do delta = 0, that is, delta = (4/(m+n))? +4m/(m+n)+m？ /4=0
That is 16/(m+n)? +4m/(m+n)+m？ /4=0
Because here m+n≠0, the above n=-m is irrelevant. Only mn=4.
Have 16/(m+4/m)? +4m/(m+4/m)+m？ /4=0
Multiply by (m+4/m) at the same time? get
16+4m(m+4/m)+m？ (m+4/m)？ /4=0
16+4m？ + 16+m？ (m？ +8+ 16/ m? )/4=0
32+4m？ +m^4/4+2m？ +4=0
m^4/4+6m？ +36=0
m^4+24m？ + 144=0
(m？ + 12)? =0
m？ =- 12
So there is no solution.
Know that this p coordinate does not exist.