find the focus and equation of parabola whose vertex is (6,-3) and diretrix is 3x - 5y + 1 = - Maths - Conic Sections

Question

find the focus and equation of parabola whose vertex is (6,-3)
and diretrix is 3x - 5y + 1 = 0

Gursheen kaur. · Accepted Answer

Equation of directrix=3x - 5y + 1 = 0--------(i)

Let the coordinate of focus = (a,b)

$∵$ the axis of parabola is perpendicular to directrix.

So the equation of axis of parabola may be taken as x+y+k=0

$∵$ it passes through (6,-3)

⇒ 6-3+k=0

⇒ k=-3

⇒ the equation of axis of parabola is x+y-3=0---------------(ii)

Now, for the point of intersection of directrix and axis of parabola

On solving (i) and (ii), we get,

$y = \frac{5}{4}$

put the value of y in (i), we get, $x = \frac{7}{4}$

Thus, the point of intersection of directrix and axis of parabola $(x, y) = (\frac{7}{4}, \frac{5}{4})$

Coordinates of $z = (\frac{7}{4}, \frac{5}{4})$

$∵$ Vertex A (6,-3) is the mid-point of focus and $z = (\frac{7}{4}, \frac{5}{4})$

$\Rightarrow 6 = \frac{a + \frac{7}{4}}{2} a n d \Rightarrow - 3 = \frac{b + \frac{5}{4}}{2} \Rightarrow 12 = a + \frac{7}{4} \Rightarrow - 6 = b + \frac{5}{4} \Rightarrow a = \frac{77}{4} \Rightarrow b = \frac{- 29}{4}$

Coordinate of focus = $(\frac{77}{4}, \frac{- 29}{4})$

$\Rightarrow \sqrt{{(x - \frac{77}{4})}^{2} + {(y - \frac{- 29}{4})}^{2}} = | \frac{3 x - 5 y + 1}{\sqrt{3^{2} + 5^{2}}} | \Rightarrow \sqrt{{(x - \frac{77}{4})}^{2} + {(y + \frac{29}{4})}^{2}} = | \frac{3 x - 5 y + 1}{\sqrt{34}} | s q u a r i n g b o t h s i d e s, w e g e t, \Rightarrow {(x - \frac{77}{4})}^{2} + {(y + \frac{29}{4})}^{2} = \frac{{(3 x - 5 y + 1)}^{2}}{34} o n s o l v i n g t h i s e q u a t i o n, w e g e t, \Rightarrow 25 x^{2} + 9 y^{2} - \frac{89}{2} x + \frac{49}{2} y + 30 x y + \frac{17}{2} [{(77)}^{2} + {(29)}^{2}] = 0 \Rightarrow 25 x^{2} + 9 y^{2} - \frac{89}{2} x + \frac{49}{2} y + 30 x y + \frac{17}{2} [5929 + 841] = 0 \Rightarrow 25 x^{2} + 9 y^{2} - \frac{89}{2} x + \frac{49}{2} y + 30 x y + \frac{17}{2} [6770] = 0 \Rightarrow 25 x^{2} + 9 y^{2} - \frac{89}{2} x + \frac{49}{2} y + 30 x y + \frac{17}{2} [6770] = 0 \Rightarrow 25 x^{2} + 9 y^{2} - \frac{89}{2} x + \frac{49}{2} y + 30 x y + 17 (3385) = 0$

which is the equation of parabola

-3

find the focus and equation of parabola whose vertex is (6,-3) and diretrix is 3x - 5y + 1 = 0.