Question

12.
The equation of the parabola where focus is
(-3,2) and directrix is x+y=4 is
a) x2 + y2 + 3xy - 2y + 1 = 0
b) x2 + y2 – 2x + 4y + 10 = 0
c) x2 + y2 – 3xy + 10x = 0
@) x2 + y2 – 2xy + 20x + 10 = 0​

Answer 1

Answer:

The equation of the parabola is

$\bf\,x^2-2xy+y^2+20x+10=0$

option (d) is correct

Step-by-step explanation:

$\text{Let p(x,y) be any point on the parabola}$

$\textbf{Given:}$

$\text{Focus S(-3,2)}$

$\text{Directrix is x+y-4=0}$

By definition of parabola

$\text{Eccentricity = \frac{SP}{PM} }$

$1=\frac{SP}{PM}$

$\implies\,SP=PM$

$\implies\sqrt{(x+3)^2+(y-2)^2}=(\frac{x+y-4}{\sqrt{1+1}})$

$\implies\sqrt{(x+3)^2+(y-2)^2}=(\frac{x+y-4}{\sqrt{2}})$

$\text{squaring on both sides}$

$(x+3)^2+(y-2)^2=(\frac{x^2+y^2+16+2xy-8y-8x}{2})}$

$\implies\;2(x^2+9+6x+y^2+4-4y)=x^2+y^2+16+2xy-8y-8x$

$\implies\;2x^2+2y^2+12x-8y+26=x^2+y^2+16+2xy-8y-8x$

$\implies\boxed{\bf\;x^2-2xy+y^2+20x+10=0}$