Question & Answer
QUESTION

The equation of parabola with focus (-1,-1) and directrix $2x - 3y + 6 = 0$ is $a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$. Then, $| a - b |$ is equal to:

ANSWER Verified

Hint: Here, equate the distance of P from focus to perpendicular distance of P from directrix. You’ll get an equation and compare the equation with the given equation to find values of a and b.

Complete step by step answer:

Let P (x,y) be a point on parabola. Then,

Distance of P from the focus = Perpendicular distance of P from the Directrix (Parabola property)

$\Rightarrow \sqrt {{{\left( {x + 1} \right)}^2} + {{\left( {y + 1} \right)}^2}} = \left| {\dfrac{{2x - 3y + 6}}{{\sqrt {{2^2} + {3^2}} }}} \right| \\$

$\Rightarrow {\left( {x + 1} \right)^2} + {\left( {y + 1} \right)^2} = \dfrac{{{{\left( {2x - 3y + 6} \right)}^2}}}{{13}} \\$

$\Rightarrow 13{x^2} + 13{y^2} + 26x + 26y + 26 = 4{x^2} + 9{y^2} + 36 - 12xy + 24x - 36y$

$\Rightarrow 9{x^2} + 4{y^2} + 12xy + 2x + 62y - 10 = 0$

So, on comparing with given equation

a = 9, b = 4, 2h = 12, 2g = 2, 2f = 62, c = - 10

$\Rightarrow \left| {a - b} \right| = \left| {9 - 4} \right| = \left| 5 \right| = 5$

NOTE: - In this particular type of question apply parabola property and solve it to get your desired answer.