Question

The equation of the parabola whose focus is $(-1,1)$ and directrix is $4x+3y-24=0$ is

• A. $9x^2+16y^2-24xy+242x+94y-526=0$
• B. $16x^2+9y^2-24xy+242x+94y-526=0$
• C. $2x^2-23y^2+7xy+32x+17y+40=0$
• D. $Noneofthese$

Answer 1

The correct option is A $9x^2+16y^2-24xy+242x+94y-526=0$

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

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

$\left|\frac{4\times -1+3\times 1-24}{\sqrt{4^2+3^2}}\right|=\left|\frac{-25}{5}\right|=5$

Hence, length of latus rectum is $10$ units.

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

${\left|\frac{4x+3y-24}{\sqrt{4^2+3^2}}\right|}^2={(x+1)}^2+{(y-1)}^2$

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

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