Question

the latus rectum of a parabola whose directrix is x +y_2 =0 and focus is 3 ,-4

Answer 1

Please see the attachment

Answer 2

The latus rectum of a parabola whose directrix is x +y_2 =0 and focus is 3 ,-4 is of length 3√2

Given the directrix equation of parabola is : x + y - 2 = 0

And its focus is (3,-4)

Putting x = 3, and y = -4 in the directrix equation, we get: 3 - 4 - 2

So, the distance between directrix and focus is:

|(3-4-2)/√2|

Again, this is equal to 2a, as it is the distance between directrix and focus.

So, |(3-4-2)/√2| = 2a

⇒ 3/√2 = 2a

⇒ a = 3/(2√2)

We know, the length of latus rectum is 4a.

So, 4a = 4[3/(2√2)] = 12/(2√2) = 6/√2

= (6√2)/2                                [Multiplying numerator and denominator by √2]

= 3√2