Answer :

Given the equation of directrix is 3x - 4y = 2 and focus is (3, 3).

We know that the directrix and axis are perpendicular to each other. The axis also passes through the focus.

Let us find the slope of the directrix.

We know that the slope of the line ax + by + c = 0 is .

⇒

⇒ .

We know that the products of the slopes of the perpendicular lines (non - vertical) is - 1. Let us assume the slope of axis is m_{2}.

⇒ m_{1}.m_{2} = - 1

⇒

⇒ .

We know that the equation of the line passing through the point (x_{1}, y_{1}) and having slope m is (y - y_{1}) = m(x - x_{1})

⇒

⇒ 3(y - 3) = - 4(x - 3)

⇒ 3y - 9 = - 4x + 12

⇒ 4x + 3y = 21

On solving the lines 4x + 3y = 21 and 3x - 4y = 2, we get the intersection point to be .

We know that the length of latus rectum is equal to the twice of the perpendicular distance between directrix and focus.

We know that the perpendicular distance from a point (x_{1}, y_{1}) to the line ax + by + c = 0 is .

⇒

⇒

⇒

⇒

⇒ L = 2

∴The length of the latus rectum is 2.

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