# Write the equation of ellipse if short axis is 2sqrt10. The point (10;-sqrt5) lies on the ellipse.

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The equation of ellipse is given by x^2/a^2 +y^2/b^2 = 1.

The shorter axis is given to be = 2sqrt10.

Therefore 2b = 2sqrt10

b = sqrt10

b^2 = 10

a^2 = b^2(1-e^2) = 10 (1-e^2).

Since (10, sqrt5) is on the ellipse, the coordinates should satisify the ellipse x^2/a^2 +y^2/10 = 1

10^2/a^2 +(sqrt5)^2/10 = 1.

100/a^2 +5/10 = 1

100/a^2 = 1-5/10 = 1/2.

a^2 = 100*2 = 200.

Substitute this value of a^2 = 200 in x^2/a^2+y^2/10 =1.

Therefore the equation of the ellipsin the standard form is x^2/200+y^2/10 = 1

The equation of the ellipse is:

x^2/a^2 + y^2/b^2 = 1

where a = long semi-axis

b = short semi-axis

To write the equation of the ellipse we'll have to calculate a, because b, the short axis, is given by the enunciation.

We also know that the ellipse passes through the given point, meaning that the coordinates of the point are verifying the equation of the ellipse.

x1^2/a^2 + y1^2/b^2 = 1

M(10,-sqrt5) belongs to ellipse if and only if:

10^2/a^2 + (-sqrt5)^2/10 = 1

100/a^2 + 5/10 = 1

We'll add -5/10 both sides:

100/a^2 = 1 - 5/10

100/a^2 = 5/10

We'll divide by 5 both sides:

20/a^2 = 1/10

We'll cross multiply:

a^2 = 200

Now, we'll write the equation of the ellipse:

x^2/a^2 + y^2/b^2 = 1

**x^2/200 + y^2/10 = 1**