Question

The equation
$25[x^2+y^2-2x+1]=(4x-3y+1{)}^2$ represents

• A. Parabola
• B. Circle
• C. Pair of straight line
• D. None of these

Answer 1

The correct option is A Parabola

The given equation is
$25[x^2+y^2-2x+1]=(4x-3y+1{)}^2\cdots (1)$
The right hand side of this equation can be written in the form of perpendicular distance
$(4x-3y+1{)}^2=25{\left(\frac{4x-3y+1}{\sqrt{(4^2+3^2)}}\right)}^2$
then equation $\left(1\right)$ can be re-written as
$25\left[\right(x-1{)}^2+(y-0{)}^2]=25{\left(\frac{4x-3y+1}{\sqrt{(4^2+3^2)}}\right)}^2$
$\Rightarrow \sqrt{(x-1{)}^2+(y-0{)}^2}=\frac{|4x-3y+1|}{\sqrt{4^2+3^2}}$
here $e=1$
Thus the given equation represents a parabola. It may noted that $(1,0)$ is the focus and $4x-3y+1=0$ is the directrix of the parabola.