Find the points of intersection of the graphs.

Question

x^2 - y^2 - 2x + 2y = 0
4x^2 + y^2 - 8x = 0
Please and thank you.

Answer 1

The equations are x² - y² - 2x + 2y = 0 and 4x² + y² - 8x = 0.

Find the points of intersection by algebraically.

Write equation 1 : x² - y² - 2x + 2y = 0 in complete square form.

(x² - 2x) - (y² - 2y) = 0

To change the expressions (x² - 2x) and (y² - 2y) into a perfect square trinomial add (half the x or y coefficient)² to each side of the expression.

 Here x coefficient = 2. so, (half the x coefficient)² = (2/2)²= 1.

Here y coefficient = 2. so, (half the y coefficient)² = (2/2)²= 1.

Add 1 and subtract 1 from each side.

(x² - 2x + 1) - (y² - 2y + 1) = 0 + 1 - 1.

(x - 1)² - (y - 1)² = 0.

(y - 1)² = (x - 1)².

y - 1= ± (x - 1)

y = ± (x - 1) + 1.

y = x and y = - x + 2.

Substitute the value of y = x in the equation 2 : 4x² + y² - 8x = 0.

4x² + x² - 8x = 0

5x² - 8x = 0

x(5x - 8) = 0

x = 0 and x = 8/5.

Substitute the value of y = -  x + 2 in the equation 2 : 4x² + y² - 8x = 0.

4x² + ( - x + 2)² - 8x = 0

4x² + x² - 4x + 4 - 8x = 0

5x² - 12x + 4 = 0

5x² - 10x  - 2x + 4 = 0

5x(x - 2) - 2(x  - 2) = 0

(5x - 2)(x  - 2) = 0

x = 2/5 and x = 2.

If x = 2/5 then y = - x + 2 = - 2/5 + 2 = (-2+10)/5 = 8/5.

If x = 2 then y = - x + 2 = - 2 + 2 = 0.

The four points of intersection are (0, 0), (8/5, 8/5), (2/5, 8/5) and (2, 0).

Answer 2

The equations are x² - y² - 2x + 2y = 0 and 4x² + y² - 8x = 0.

Write equation 1 : x² - y² - 2x + 2y = 0 in complete square form.

(x² - 2x) - (y² - 2y) = 0

To change the expressions (x² - 2x) and (y² - 2y) into a perfect square trinomial add (half the x or y coefficient)² to each side of the expression.

 Here x coefficient = 2. so, (half the x coefficient)² = (2/2)²= 1.

Here y coefficient = 2. so, (half the y coefficient)² = (2/2)²= 1.

Add 1 and subtract 1 from each side.

(x² - 2x + 1) - (y² - 2y + 1) = 0 + 1 - 1.

(x - 1)² - (y - 1)² = 0.

(y - 1)² = (x - 1)².

y - 1= ± (x - 1)

y = ± (x - 1) + 1.

y = x and y = - x + 2.

The equation 1 : 1 : x² - y² - 2x + 2y = 0 represent the two line equations, y = x and y = - x + 2.

Write equation 2 : 4x² + y² - 8x = 0 in complete square form.

4x² - 8x + y² = 0

(x² - 2x) + y²/4 = 0

(x² - 2x) + y²/4 = 0

To change the expressions (x² - 2x) and (y² - 2y) into a perfect square trinomial add (half the x or y coefficient)² to each side of the expression.

 Here x coefficient = 2. so, (half the x coefficient)² = (2/2)²= 1.

Add 1 and to each side.

(x² - 2x + 1) + y²/4 = 0 + 1

(x - 1)² + y²/4 = 1

(x - 1)²/1² + y²/2² = 1

The above equation represent the ellipse.

Compare the equation (x - 1)²/1² + (y - 0)²/2² = 1 with standard form of ellipse equation : (x + h)²/b² + (y - k)²/a² = 1 (Major axis is vertical since denominator of the y² - term (2²) > denominator of the x² - term (1²)).

a = 2, b = 1, Center = (h, k) = (1, 0), Vertices = (h, k ± a) = (1, 0 ± 2) = (1, 2) and (1, -2),

Co - vertices = (h± b, k) = (1±1, 0) = (2, 0) and (0, 0).

Find the points of intersection by graphically.

1. Draw coordinate plane.

2. Graph the two lines y = x and y = - x + 2 by using slope - intercept form.

3. To draw the ellipse, plot the center = (h, k) = (1, 0).

4. The value of a² = 2², the end points of the major axis lie 2 units up and down from the center. So, plot the Vertices of the are ellipse (1, 2) and (1, -2).

5. Similarly, the value of b² = 1², the end points of the minor axis lie 1 units right and left from the center. So, plot the Co - vertices of the are ellipse (2, 0) and (0, 0).

6. Connect these points to obtain ellipse.

7. Observe the graph, the four points of intersection are (0, 0), (8/5, 8/5), (2/5, 8/5) and (2, 0).