Question

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

• A. an ellipse
• B. a parabola
• C. a hyperbola
• D. a pair of straight lines

Answer 1

The correct option is B 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, $e=1$
The fixed point $(1,0)$ does not lie on the fixed line $4x-3y+1=0$
Thus, the given equation represents a parabola.