Find the equation of the parabola whose focus is (2,5)and dielectric is 3x 4y 1=0.?

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Answer

Finding the Equation of a Parabola Given Its Focus and Directrix

Given that the focus of a parabola is (2,5) and its directrix is 3x+4y+1=0, we can find its equation. The general equation of a parabola with a vertical axis of symmetry is:

(x - h)² = 4p(y - k)

where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix. We can find p by using the distance formula:

p = |(ax + by + c)/√(a² + b²)|

where ax + by + c = 0 is the equation of the directrix and a² + b² = 1. We can rewrite the equation of the directrix as:

4y = -3x - 1

y = (-3/4)x - 1/4

Comparing this to the standard form of a line, y = mx + b, we can see that the slope is -3/4 and the y-intercept is -1/4. Therefore, a = -3/5 and b = 4/5. Substituting these values into the formula for p, we get:

p = |(-3/5)(2) + (4/5)(5) + 1)/√((-3/5)² + (4/5)²)|

p = |(-6/5) + (20/5) + 1)/√(9/25 + 16/25)|

p = |15/5/√25/25|

p = 3

Therefore, the vertex of the parabola is (2,2) (since the focus is 3 units above the vertex) and the value of p is 3. Substituting these values into the equation of the parabola, we get:

(x - 2)² = 4(3)(y - 2)

Simplifying, we get:

(x - 2)² = 12(y - 2)

Therefore, the equation of the parabola is (x - 2)² = 12(y - 2).

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