If the slope of the straight line passing through point F 1 does not exist, that is, the linear equation is x= 1, at this time, A (1, 3/2), B (1, 3/2), AB ⊥ X axis, AB = 3, F655.
S △ PAB =1/2 * ab * f1p = 3/2.
So the slope of the straight line passing through point F 1 exists, not 0. Let the equation of straight line AB be y=k(x- 1)(k≠0), that is, kx-y-k=0, and let A (x 1, y 1), b (.
The distance of straight line AB at point P (2,0) is
d=|2k-0-k|/√(k? + 1)=|k|/√(k? + 1)
By y=kx-k, 3x? +4y? = 12
get
(4k? +3)x? -8k? x+4k? - 12=0
x 1+x2=8k? /(4k? +3)
x 1x2=(4k? - 12)/(4k? +3)
(x 1+x2)? -4x 1x2
=[64k^4-( 16k? -48)(4k? +3)]/(4k? +3)?
=( 144k? + 144)/(4k? +3)?
AB=√( 1+k? )*√[(x 1+x2)? -4x 1x2]
= 12(k? + 1)/(4k? +3)
S△PAB= 1/2*AB*d
that is
6(k? + 1)/(4k? +3)*|k|/√(k? + 1)=36/ 13
|k|*√(k? + 1)/(4k? +3)=6/ 13
k? (k? + 1)/(4k? +3)? =36/ 169
Let x=k? (x≥0),
x(x+ 1)/(4x+3)? =36/ 169
36( 16x? +24x+9)= 169x? + 169
407x? +864x+ 155=0
x= 1.93
So k=√x= 1.4.
The linear equation is y= 1.4x- 1.4.