Question

If focus of a parabola is (1, 2) and feet of perpendiculars on any two tangents to this parabola from focus are (3, 4) and (4, 6). Then vertex of the parabola is

Answer 1

Given info : If focus of a parabola is (1,2) and feet of perpendicular on any two tangents to this parabola from focus are (3 , 4) and (4 , 6).

To find : the vertex of the parabola is...

Solution : equation of line joining two points (3, 4) and (4,6) is ...

(y - 4) = (6 - 4)/(4 - 3) (x - 3)

⇒y - 4 = 2(x - 3)

⇒y - 4 = 2x - 6

⇒2x - y = 2 ......(1)

now slope of line passing through focus = -1/slope of line joining (3,4) and (4,6)

[Because both are perpendicular ].

= -1/2

Now equation of line passing through focus (1,2) is (y - 2) = -1/2(x - 1)

⇒2y - 4 + x - 1 = 0

⇒x + 2y = 5 .......(2)

from equations (1) and (2) we get,

x = 9/5 and y = 8/5

Therefore the vertex of parabola is (9/5, 8/5)