# The equation of the parabola whose focus is ( 3,-4 ) and directrix 6x- 7y + 5=0 is;

$

(a)\;{(7x + 6y)^2} - 570x + 750y + 2100 = 0 \\

(b)\;{(7x + 6y)^2} + 570x - 750y + 2100 = 0 \\

(c)\;{(7x - 6y)^2} - 570x + 750y + 2100 = 0 \\

(d)\;{(7x - 6y)^2} + 570x - 750y + 2100 = 0 \\

$

Answer

Verified

364.2k+ views

Hint: For any point on the line of parabola, the distance to the focus is equal to the perpendicular distance to the directrix.

We know that for any point P(x, y) on the line parabola, the distance to the focus is F(3,-4) is equal to the perpendicular distance to the Directrix line d is,

6x - 7y + 5=0

$ \Rightarrow \frac{{{{\left( {6x - 7y + 5} \right)}^2}}}{{\left( {{6^2} + {7^2}} \right)}} = {(x - 3)^2} + {\left( {y + 4} \right)^2}$

Now we know that $\left[ {{{(a + b + c)}^2} = {a^2} + {b^2} + {c^2} + 2(ab + bc + ca)} \right]$ and hence on applying the same formula we have,

$ \Rightarrow 36{x^2} + 49{y^2} + 25 - 84xy - 70y + 60x = 85{x^2} + 85{y^2} - 510x + 2125 + 680y$

And hence on doing the simplification, we have

$ \Rightarrow 49{x^2} + 36{y^2} + 84xy - 570x + 750y + 2100 = 0$

And hence it can be written as,

$ \Rightarrow \;{(7x + 6y)^2} - 570x + 750y + 2100 = 0$

So option a is correct answer.

Note: In this type of question first of all we have to find the directrix as well as the Distance to the focus and with the help of that we can find the equation of Parabola.

We know that for any point P(x, y) on the line parabola, the distance to the focus is F(3,-4) is equal to the perpendicular distance to the Directrix line d is,

6x - 7y + 5=0

$ \Rightarrow \frac{{{{\left( {6x - 7y + 5} \right)}^2}}}{{\left( {{6^2} + {7^2}} \right)}} = {(x - 3)^2} + {\left( {y + 4} \right)^2}$

Now we know that $\left[ {{{(a + b + c)}^2} = {a^2} + {b^2} + {c^2} + 2(ab + bc + ca)} \right]$ and hence on applying the same formula we have,

$ \Rightarrow 36{x^2} + 49{y^2} + 25 - 84xy - 70y + 60x = 85{x^2} + 85{y^2} - 510x + 2125 + 680y$

And hence on doing the simplification, we have

$ \Rightarrow 49{x^2} + 36{y^2} + 84xy - 570x + 750y + 2100 = 0$

And hence it can be written as,

$ \Rightarrow \;{(7x + 6y)^2} - 570x + 750y + 2100 = 0$

So option a is correct answer.

Note: In this type of question first of all we have to find the directrix as well as the Distance to the focus and with the help of that we can find the equation of Parabola.

Last updated date: 25th Sep 2023

â€¢

Total views: 364.2k

â€¢

Views today: 11.64k

Recently Updated Pages

What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the past tense of read class 10 english CBSE