Question

What is the length of the latus rectum of the parabola whose focus is `(-1, 1)` and directrix is `4x+3y-24=0`?

Answer 1

Length of latus rectum is `10`.

Explanation:

Length of latus rectum is twice the distance between focus and vertex or four times the distance of focus from directrix.

The distance of focus `(-1,1)` from directrix `4x+3y-24=0` is

`|(4xx(-1)+3xx1-24)/sqrt(4^2+3^2)|`

= `|(-4+3-24)/5|=|-25/5|=5`

Hence, length of latus rectum is `10`.

As parabola is locus of a point, which moves so that its distance from focus and directrix is alwaays equal, its equation is

`|(4x+3y-24)/sqrt(4^2+3^2)|^2=(x+1)^2+(y-1)^2`

or `16x^2+9y^2+576+24xy-192x-144y=25x^2+50x+25+25y^2-50y+25`

or `9x^2+16y^2-24xy+242x+94y-526=0`

Ends of latus rectum are `(-4,5)` and `(2,-3)`

graph{(9x^2+16y^2-24xy+242x+94y-526)((x+1)^2+(y-1)^2-0.08)((x+4)^2+(y-5)^2-0.08)((x-2)^2+(y+3)^2-0.08)(4x+3y-24)(4x+3y+1)=0 [-10.42, 9.58, -4.16, 5.84]}