Question

Let 3x-y-8=0 be the equation of tangent to a parabola at the point (7,13). If the focus of the
parabola is at (-1,-1). Its directrix is​

Answer 1

Answer:

dauragbn asmd

Step-by-step explanation:

3x y 8 mp

wiiwueeje8eieje

Answer 2

The equation of directrix is x+8y+19=0.

Given:

Equation of tangent to a parabola at point (7,13) i.e. 3x-y-8 = 0

Coordinates of focus of parabola = (-1,-1)

To Find:

The directrix of the parabola

Solution:

We can simply solve this problem by using the following mathematical process.

The image of the focus along any tangent lies on the directrix.

Let,

The coordinates of the image = (h,k)

The line joining the focus and image will form an angle of 90° with the equation given.

Now, the slope of the line joining focus and image will be given by the formula $\frac{y_{2} -y_{1} }{x_{2} -x_{1} }$

Using the formula we get,

Slope = $\frac{k+1}{h+1}$

Now, using the slopes of perpendicular lines $m_{1} m_{2} =-1$

Putting the values of $m_{2} and m_{1}$

$(\frac{k+1}{h+1}) (3)=-1$

$3k+3=-h-1$     (i)

Since the equation of the tangent will cut the line joining the focus and image in half, let the coordinates at the intersection be (a,b)

Then,

$a=\frac{h-1}{2}$  and  $b=\frac{k-1}{2}$

The points a and b should satisfy the equation of the tangent.

∴ $\frac{k-1}{2} =\frac{3(h-1)}{2} -8$

$k-1=3h-19$

Using (i)

$k-1=3(-3k-4)-19$

$k=-3$ and $h=5$

Now, the slope of coordinates of the image and the point given i.e. (7,13) = -1 ÷ Slope of directrix

-1 ÷ Slope of directrix = $\frac{13-(-3)}{7-5}$

The slope of directrix = $\frac{-1}{8}$

∴ Equation of directrix

$y+3=\frac{-1}{8} (x-5)$

$x+8y+19=0$

Hence, the equation of directrix is x+8y+19=0.

#SPJ3