Find the equation of the parabola whose focus is the point (2, 3) and directrix is the line x – 4y + 3 = 0. Also, find the length of its latus - rectum.
Given that we need to find the equation of the parabola whose focus is S(2, 3) and directrix(M) is x - 4y + 3 = 0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x1, y1) and (x2, y2) is 
.
We know that the perpendicular distance from a point (x1, y1) to the line ax + by + c = 0 is 
.
⇒ SP = PM
⇒ SP2 = PM2
⇒ 
⇒ 
⇒ 
⇒ 17x2 + 17y2 - 68x - 102y + 221 = x2 + 16y2 + 6x - 24y - 8xy + 9
⇒ 16x2 + y2 + 8xy - 74x - 78y + 212 = 0
∴The equation of the parabola is 16x2 + y2 + 8xy - 74x - 78y + 212 = 0.
We know that the length of the latus rectum is twice the perpendicular distance from the focus to the directrix.
⇒ 
⇒ 
⇒ 
∴The length of the latus rectum is 
.