Mathematics: Given points A(-1, -2) and B(2, 4) where AP: BP=1:2, find the locus of point P.

70000Step-by-step explanation:...

Given points A(-1, -2) and B(2, 4) where AP:

Ответ:

jhonpiper

05.08.2021

Mathematics

x^2 + 4x + y^2 +8y = 0

Step-by-step explanation:

Given

A = (-1,-2)

B = (2,4)

AP:BP = 1 : 2

Required

The locus of P

AP:BP = 1 : 2

Express as fraction

$\frac{AP}{BP} = \frac{1}{2}$

Cross multiply

2AP = BP

Calculate AP and BP using the following distance formula:

$d = \sqrt{(x - x_1)^2 + (y - y_1)^2}$

So, we have:

$2 * \sqrt{(x - -1)^2 + (y - -2)^2} = \sqrt{(x - 2)^2 + (y - 4)^2}$

$2 * \sqrt{(x +1)^2 + (y +2)^2} = \sqrt{(x - 2)^2 + (y - 4)^2}$

Take square of both sides

4 * [(x +1)^2 + (y +2)^2] = (x - 2)^2 + (y - 4)^2

Evaluate all squares

4 * [x^2 + 2x + 1 + y^2 +4y + 4] = x^2 - 4x + 4 + y^2 - 8y + 16

Collect and evaluate like terms

4 * [x^2 + 2x + y^2 +4y + 5] = x^2 - 4x + y^2 - 8y + 20

Open brackets

4x^2 + 8x + 4y^2 +16y + 20 = x^2 - 4x + y^2 - 8y + 20

Collect like terms

4x^2 - x^2 + 8x + 4x + 4y^2 -y^2 +16y + 8y + 20 - 20 = 0

3x^2 + 12x + 3y^2 +24y = 0

Divide through by 3

x^2 + 4x + y^2 +8y = 0

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Given points A(-1, -2) and B(2, 4) where AP: BP=1:2, find the locus of point P.

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Given points A(-1, -2) and B(2, 4) where AP: BP=1:2, find the locus of point P.​

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Given points A(-1, -2) and B(2, 4) where AP: BP=1:2, find the locus of point P.