Question

the equation of directrix of parabola which touches x axis and y axis at (4,0) and (2,0) resp is

Answer 1

Answer:

Given in the question, Parabola touches X-axis at (1,0) and Y-axis at (0,2).

Let the equation of directrix is parabola y=mx and co-ordinate focus of parabola is (h,k).

We know that, Equation of parabola is

\sqrt{(x-h)^{2} +(y-k)^{2} } = / \frac{y-mx}{\sqrt{1+m^2} }} } /

Taking squares on both Sides, we get

{(x-h)^{2} +(y-k)^{2} } = / \frac{(y-mx)^2}{(1+m)^{2} } /

If passes through the point (1,0) Then,

(0-h)^{2} + (1-k)^{2} = \frac{1}{1+m^2} .............. equation 1

It also passes through (0,2)

(2-h)^{2} + k^2 = \frac{m^2}{1+m^2} ................. equation 2

Adding equation 1 and 2, we have

h^2 + (1-k)^{2} + (2-h)^{2} + k^2 = \frac{1}{1+m^2} + \frac{m^2}{1+m^2}

2h^2 + 2k^2 - 4h - 2k + 5 = 1

2h^2 + 2k^2 - 4h - 2k + 4 = 0

h^2 + k^2 - 2h - k + 2 = 0

Hence the locus of (x, y)