The equation of the parabola with focus (a, b | vedclass

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Question

(A) The definition of a parabola is the locus of a point $P(x, y)$ such that its distance from the focus $(a, b)$ is equal to its perpendicular distan

Vedclass · Accepted Answer

$(ax - by)^2 - 2a^3x - 2b^3y + a^4 + a^2b^2 + b^4 = 0$

Vedclass · Answer

(A) The definition of a parabola is the locus of a point $P(x, y)$ such that its distance from the focus $(a, b)$ is equal to its perpendicular distance from the directrix $\frac{x}{a} + \frac{y}{b} - 1 = 0$.$(x - a)^2 + (y - b)^2 = \left( \frac{\frac{x}{a} + \frac{y}{b} - 1}{\sqrt{(\frac{1}{a})^2 + (\frac{1}{b})^2}} \right)^2$$(x - a)^2 + (y - b)^2 = \left( \frac{bx + ay - ab}{\sqrt{a^2 + b^2}} \right)^2$$(x^2 - 2ax + a^2 + y^2 - 2by + b^2) = \frac{(bx + ay - ab)^2}{a^2 + b^2}$$(a^2 + b^2)(x^2 + y^2 - 2ax - 2by + a^2 + b^2) = (bx + ay - ab)^2$Expanding both sides and simplifying leads to:$(ax - by)^2 - 2a^3x - 2b^3y + a^4 + a^2b^2 + b^4 = 0$.

The equation of the parabola with focus $(a, b)$ and directrix $\frac{x}{a} + \frac{y}{b} = 1$ is given by

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