Question

The equation of the parabola whose focus is the point (0, 0) and the tangent at the vertex is x – y + 1 = 0 is

• A. $x^2+y^2$ – 2xy – 4x – 4y – 4 = 0
• B. $x^2+y^2$ – 2xy + 4x – 4y – 4 = 0
• C. $x^2+y^2$ + 2xy – 4x + 4y – 4 = 0
• D. $x^2+y^2$ + 2xy – 4x – 4y + 4 = 0

Answer 1

The correct option is C $x^2+y^2$ + 2xy – 4x + 4y – 4 = 0

Tangent at vertex is x – y + 1 = 0 and focus (0, 0)
$\therefore$ Equation of axis of the parabola is x + y = 0
$\therefore vertex=(-\frac{1}{2},\frac{1}{2})$
Let equation of directrix be x – y + k = 0 and it passes through (–1, 1)
$\therefore$ k = 2
Directrix $\equiv$ x – y + 2 = 0
$\therefore$ Equation of parabola
${\left(\frac{x-y+2}{\sqrt{2}}\right)}^2=x^2+y^2\Rightarrow x^2+y^2+2xy-4x+4y-4=0$