Question

Write the coordinates the vertex of the parabola whose focus is at (-2,1) and directrix is the line x+y-3=0.

Answer 1

The equation of directrix is x+y-3=0 ...(i)

$\Rightarrow y=-x+3$

$\therefore \text{slope of line}=m_1=-1$

$\therefore \text{slope of line}=m_2=\frac{-1}{m_1}=\frac{-1}{-1}=1$

The equation of line posses through (2,-1) with slope 1 is

$y-1=1[x-(-2\left)\right][\because y-y_0=m(x-x_0\left)\right]$

$\Rightarrow y-1=x+2$

$\Rightarrow y-x=3...\left(ii\right)$

$y-x-3=0$

Adding equations (i) and (ii),we get

$2y-6=0$

$\Rightarrow 2y=6$

$\Rightarrow y=\frac{6}{2}=3$

Putting y=3 in equation (i), we get

$x+3-3=0$

$\Rightarrow x=0$

$\therefore$ (0,3) be the coordinates of the point of intersection of the axis and directrix.

Then,coordinates of vertex $(x_1,y_1)$ is the mid-point of the line segment joining (0,3) and (-2,1)

$\therefore \text{required coordinates of vertex are}(-1,2)$

Then,we have :

$\Rightarrow (x_1,y_1)=(-1,2)$