Question

how to find the coordinates of the intersection here? I know how to do it with a line and a circle but not sure about a line and an ellipse

Answer 1

Answer: (3.2, 1.8)

3.2 = 16/5 and 1.8 = 9/5

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Work Shown

`\text{x} + \text{y} = 5\\\\\text{y} = 5 - \text{x}\\\\\frac{\text{x}^2}{16}+\frac{\text{y}^2}{9} = 1\\\\144*\left(\frac{\text{x}^2}{16}+\frac{\text{y}^2}{9}\right) = 144*1\\\\9\text{x}^2+16\text{y}^2 = 144\\\\9\text{x}^2+16(5-\text{x})^2 = 144\\\\`

Let's solve for x.

`9\text{x}^2+16(5-\text{x})^2 = 144\\\\9\text{x}^2+16(25-10\text{x}+\text{x}^2) = 144\\\\9\text{x}^2+400-160\text{x}+16\text{x}^2 = 144\\\\25\text{x}^2-160\text{x}+256 = 0\\\\`

We'll need to use the quadratic formula from here.

Plug in a = 25, b = -160, c = 256.

`\text{x} = (-b\pm√(b^2-4ac))/(2a)\\\\\text{x} = (-(-160)\pm√((-160)^2-4(25)(256)))/(2(25))\\\\\text{x} = (160\pm√(25600 - 25600))/(50)\\\\\text{x} = (160\pm√(0))/(50)\\\\\text{x} = (160\pm0)/(50)\\\\\text{x} = (160)/(50)\\\\\text{x} = (16)/(5)\\\\\text{x} = 3.2\\\\`

Then use this x value to find its paired y value.

y = 5 - x

y = 5 - 3.2

y = 1.8

The point of intersection is (3.2, 1.8)

GeoGebra can be used to verify the answer. Desmos is another good option. If you prefer something like a TI83, then stick to that.

The decimal values 3.2 and 1.8 convert to the improper fractions 16/5 and 9/5 in that order.

Because we have one point of intersection, the line is tangent to the ellipse.