Question

Trace the conic 9x2 + 24xy + 16y2 - 2x + 14y + 1 = 0 and find the
coordinates of its focus and the equation of directrix.
​

Answer 1

Answer:

Trace the conic 9x2 + 24xy + 16y2 - 2x + 14y + 1 = 0 and find the

coordinates of its focus and the equation of directrix.

Answer 2

We need to recall the following properties of the parabola.

Standard equation: $y^{2}=4ax$

Focus: $(a,0)$

Equation of directrix: $x+a=0$

Given:

$9x^{2} +24xy+16y^2-2x+14y+1=0$

Rewrite the equation as follows.

$9x^{2} +24xy+16y^2=2x-14y-1$

$(3x+4y)^2=2x-14y-1$

$(\frac{(3x+4y)}{\sqrt{25} } )^2=\frac{\sqrt{17} }{\sqrt{25}\sqrt{25} } (\frac{2x-14y-1}{\sqrt{17} } )$

This equation is in form of a standard equation of a parabola $Y^2=4aX$.

Here,

$4a=\frac{\sqrt{17} }{25}$  ,  $Y=\frac{3x+4y}{\sqrt{25} }$   and  $X=\frac{2x-14y-1}{\sqrt{17} }$

This conic is a parabola.

The equation of directrix for this parabola is $X=-a$.

$\frac{2x-14y-1}{\sqrt{17} }=-\frac{\sqrt{17} }{100}$

$200x-1400y-100=-17$

$200x-1400y-83=0$

Thus, the equation of directrix of a parabola is $200x-1400y-83=0$ .

For coordinates of a focus:

x-coordinate is: $X=a$

$\frac{2x-14y-1}{\sqrt{17} }=\frac{\sqrt{17} }{100}$

$200x-1400y-100=17$

$200x-1400y-117=0$                   .......$(1)$

y-coordinate is: $Y=0$

$\frac{3x+4y}{\sqrt{25} }=0$

$3x+4y=0$                                    ........$(2)$

Solving the equations $(1)$ and $(2)$ , we get

$x=\frac{117}{125}$  and  $y=\frac{-351}{5000}$

Hence, the coordinates of its focus are $(\frac{117}{125}, \frac{-351}{5000} )$