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Given F(0, 8) and d: y = 25/2. The distance from P(x, y) to F is 4/5 the distanc...

Oct 1, 2024

Given F(0, 8) and d: y = 25/2. The distance from P(x, y) to F is 4/5 the distance from P to d. Write an equation to show that P lies on an ellipse.

Generated Graph

Solution by Steps

step 1

Given points F(0, 8) and line d: y = 25/2, we need to find the distances. The distance from point P(x, y) to F is given by:

d(P, F) = \sqrt{(x - 0)^2 + (y - 8)^2}

step 2

The distance from point P to line d is calculated as:

d(P, d) = |y - 25/2| / \sqrt{1^2} = |y - 25/2|

step 3

According to the problem, the distance from P to F is

\frac{4}{5}

the distance from P to d, leading to the equation:

\sqrt{x^2 + (y - 8)^2} = \frac{4}{5} |y - 25/2|

step 4

Squaring both sides gives:

x^2 + (y - 8)^2 = \left(\frac{4}{5}\right)^2 (y - 25/2)^2

step 5

Expanding both sides results in:

x^2 + (y^2 - 16y + 64) = \frac{16}{25}(y^2 - 25y + 625/4)

step 6

Rearranging the equation leads to the standard form of an ellipse:

4x^2 + 25y^2 - 400y + 1600 = 0

Answer

The equation of the ellipse is

4x^2 + 25y^2 - 400y + 1600 = 0

Key Concept

The relationship between distances in an ellipse is defined by the ratio of distances from a point to the foci and the directrix.

Explanation

The derived equation represents the geometric locus of points (P) that maintain a specific ratio of distances to the focus and the directrix, characteristic of an ellipse.