Question

(a) If the equation $\lambda x^{2}4xy+y^2+\lambda x+3y+2=0$ represents a parabola then find

(b) Find the length of latus rectum of the parabola $169\{(x-1{)}^2+{\left(y-3\right)}^2\}=(5x-12y+17{)}^2.$

Answer 1

Length of latus rectum of parabola is generally given by : $L.R=4a$, where :

$a=\frac{1}{2}$ ( Distance between focus and Directrix)

Now equation of parabola is given as:

$(x-1{)}^2+(y-3{)}^2={\left(\frac{5x-12y+17}{13}\right)}^2$

This means the distance of point $1,3$ is same as its perpendicular distance from line $5x-12y+17$ which means its focus lies at $(1,3)$

and equation of directrix : $5x-12y+17$

$a=\frac{1}{2}\left|\frac{5\left(1\right)-12\left(3\right)+17}{13}\right|$

$=\frac{1}{2}\left|\frac{-14}{13}\right|$

$a=\frac{7}{13}$

$\therefore$ Length of latus rectum, $LR=4a$

$=\frac{4\times 7}{13}$

$=\frac{28}{13}$