Question

Coordinates of parametric point on the parabola, whose focus is $(\frac{-3}{2},-3)$ and the directrix is $2x+5=0$ is given by

• A. $(2t^2+2,2t-3)$
• B. $(\frac{1}{2}{t}^2-2,t-3)$
• C. $(\frac{1}{2}{t}^2-2,t+3)$
• D. $(\frac{1}{2}{t}^2+2,t+3)$

Answer 1

The correct option is B $(\frac{1}{2}{t}^2-2,t-3)$

${(x+\frac{3}{2})}^2+(y+3{)}^2={\left(\frac{2x+5}{2}\right)}^2\Rightarrow 4[x^2+\frac{9}{4}+3x]+4[y^2+9+6y]=(4x^2+25+20x)\Rightarrow (y^2+6y+9)-2(x+2)=0$
or $(y+3{)}^2=2(x+2)$
Here $a=\frac{1}{2}$
Therefore $x=\frac{1}{2}{t}^2-2$ and $y=t-3$ satisfy it for all $t$.