Could someone explain to me how to solve the following problem?

Question

Show that the ellipse 4x^2+9y^2=45 and the hyperbola x^2-4y^2=5 intersect at right angles.

Answer 1

x = ±3, y = ±1
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Ideas:
4x^2+9y^2=45 ......(1)
x^2-4y^2=5 ......(2)
(1) - 4*(2): 25y^2 = 25 => y = ±1
x^ = 5+4y^2 = 9 => x = ± 3
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To show the two equations intersect at right angles at the four points, you can show that the product of two slopes (m1 and m2) is -1 at each point.
Differentiate (1) and (2) with respect to x,
8x+18yy' = 0 =>y' = m1 = -4x/9
2x-8yy' = 0 => y' = m2 = x/4
m1*m2 = -x^2/9 = -1 at x = ±3

Answer 2

First find the points of intersection. From the equation for the hyperbola, we find x^2 = 5 + 4y^2. Putting this value of x^2 into the equation for the ellipse, we find y^2 = 1 so y = +/-1. Using these values for y in either equation we find x = +/-3. The four points of intersection are (3,1), (3,-1), (-3,1), and
(-3,-1).

Now we find the slopes of the tangents to the two curves. Using inplicit differentiation on the hyperbola we find 2x -8yy' = 0, so y' = x/4y. Doing the same for the ellipse we get
8x + 18yy' = 0, or y' = -4x/9y.

At (3,1), the ellipse has slope -12/9 = -4/3, and the hyperbola has slope 3/4. Since the product of these slopes is -1, the tangents (and hence the curves) are perpendicular.

Do the same at the other three points of intersection.

Answer 3

If you create a system of two equation, the results will show you were the two curves actually intersect each other.
1) 4x^2+9y^2=45
2) x^2-4y^2=5

2)x^2=4y^2+5
1)4(4y^2+5)+9y^2=45

1)25y^2=25 therefore y=+/- 1
2)x^2=9 therefore x=+/- 3

Answer 4

4x² + 9y² = 45
x² - 4y² = 5

solving simultaneously

4x² + 9y² = 45
4x² - 16y² = 20

25y² = 25 ...... y = ± 1 , x = ±3 are the points of intersection

differentiating to find the gradient

ellipse dy/dx = - 4x /9y ...... hyperbola dy/dx = x/4y

(3,1) dy/dx = -4/3 ................................. dy/dx = 3/4

(3, -1) dy/dx = 4/3 ................................... dy/dx = -3/4

(-3, 1) dy/dx = 4/3 ................................... dy/dx = -3/4

(-3,-1) dy/dx = -4/3 .................................. dy/dx = 3/4

You can see that the product of the gradients at each point is -1 so they intersect at right angles.