Question

find the equation of the parabola if the focus is (-3,2) and directrix is x+y=4.

Answer 1

Answer:

Step-by-step explanation:

Given the focus of the parabola $\tt{S(-3,2)}$

Directrix : $\tt{x+y-4=0}$

Let $\tt{P(x,y)}$ be any point on the parabola and let PM be the length of the perpendicular from P to the directrix

So,

$\sf{SP=PM}$

$\sf{\implies\,\sqrt{(x+3)^2+(y-2)^2}=\dfrac{|x+y-4|}{\sqrt{(1)^2+(1)^2}}}$

$\sf{\implies\,\sqrt{(x+3)^2+(y-2)^2}=\dfrac{|x+y-4|}{\sqrt{2}}}$

$\sf{\implies\,(x+3)^2+(y-2)^2=\left(\dfrac{|x+y-4|}{\sqrt{2}}\right)^{2}}$

$\sf{\implies\,x^2+6x+9+y^2-4y+4=\dfrac{|x+y-4|^2}{2}}$

$\sf{\implies\,x^2+y^2+6x-4y+13=\dfrac{x^2+y^2+2xy-8y-8x+16}{2}}$

$\sf{\implies\,2x^2+2y^2+12x-8y+26=x^2+y^2+2xy-8y-8x+16}$

$\sf{\implies\,x^2+y^2-2xy+20x+10=0}$