1. Tardigrade
  2. Question
  3. Mathematics
  4. If the equation of the parabola, whose vertex is at (5,4) and the directrix is 3 x+y-29=0, is x2+a y2+b x y+c x+d y+k=0 then a+b+c+d+k is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If the equation of the parabola, whose vertex is at $(5,4)$ and the directrix is $3 x+y-29=0$, is $x^{2}+a y^{2}+b x y+c x+d y+k=0$ then $a+b+c+d+k$ is equal to

JEE MainJEE Main 2022Conic Sections

Solution:

Vertex $(5,4)$
Directrix : $3 x+y-29=0$
Co-ordinates of $B$ (foot of directrix)
$\frac{x-5}{3}=\frac{y-4}{1}=-\left(\frac{15+4-29}{10}\right)=1$
image
$x=8, y=5$
$S=(2,3)$ (focus)
Equation of parabola
$P S=P M$
so equation is
$x^{2}+9 y^{2}-6 x y+134 x-2 y-711=0$
$a+b+c+d+k=9-6+134-2-711=-576$