Sure, let's work through the problem step-by-step:
### Part (a): Find the coordinates of the points [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
We are given:
1. The line equation: [tex]\( y = 2 - x \)[/tex]
2. The curve equation: [tex]\( 5x^2 - y^2 = 20 \)[/tex]
First, we will substitute the line equation [tex]\( y = 2 - x \)[/tex] into the curve equation [tex]\( 5x^2 - y^2 = 20 \)[/tex].
1. Substitute [tex]\( y = 2 - x \)[/tex] into [tex]\( 5x^2 - y^2 = 20 \)[/tex]:
[tex]\[
5x^2 - (2 - x)^2 = 20
\][/tex]
2. Expand and simplify [tex]\( (2 - x)^2 \)[/tex]:
[tex]\[
(2 - x)^2 = 4 - 4x + x^2
\][/tex]
3. Substitute this back in the equation:
[tex]\[
5x^2 - (4 - 4x + x^2) = 20
\][/tex]
4. Simplify the equation:
[tex]\[
5x^2 - 4 + 4x - x^2 = 20
\][/tex]
[tex]\[
4x^2 + 4x - 4 = 20
\][/tex]
[tex]\[
4x^2 + 4x - 24 = 0
\][/tex]
[tex]\[
x^2 + x - 6 = 0
\][/tex]
5. Solve for [tex]\( x \)[/tex] using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[
a = 1, \quad b = 1, \quad c = -6
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-6)}}{2(1)}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{1 + 24}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm \sqrt{25}}{2}
\][/tex]
[tex]\[
x = \frac{-1 \pm 5}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x = \frac{4}{2} = 2 \quad \text{or} \quad x = \frac{-6}{2} = -3
\][/tex]
6. Corresponding [tex]\( y \)[/tex] values using [tex]\( y = 2 - x \)[/tex]:
[tex]\[
\text{When } x = 2, \quad y = 2 - 2 = 0
\][/tex]
[tex]\[
\text{When } x = -3, \quad y = 2 - (-3) = 5
\][/tex]
Thus, the coordinates of the points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are:
[tex]\[
A(-3, 5) \quad \text{and} \quad B(2, 0)
\][/tex]
### Part (b): Find the length of the line [tex]\( AB \)[/tex].
To find the length of the line segment [tex]\( AB \)[/tex] between these two points, we use the Euclidean distance formula:
[tex]\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substitute the coordinates [tex]\( A(-3, 5) \)[/tex] and [tex]\( B(2, 0) \)[/tex]:
[tex]\[
AB = \sqrt{(2 - (-3))^2 + (0 - 5)^2}
\][/tex]
[tex]\[
AB = \sqrt{(2 + 3)^2 + (-5)^2}
\][/tex]
[tex]\[
AB = \sqrt{5^2 + (-5)^2}
\][/tex]
[tex]\[
AB = \sqrt{25 + 25}
\][/tex]
[tex]\[
AB = \sqrt{50} = 5\sqrt{2}
\][/tex]
Thus, the length of the line [tex]\( AB \)[/tex] is:
[tex]\[
5\sqrt{2}
\][/tex]
In summary:
a. The coordinates of points [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are [tex]\((-3, 5)\)[/tex] and [tex]\((2, 0)\)[/tex] respectively.
b. The length of the line [tex]\( AB \)[/tex] is [tex]\( 5\sqrt{2} \)[/tex].
