show that the line x+y=5 touches the circle x²+y²-2x-4y+3=0.also, find point of the contact.
To show that the line x+y=5 touches the circle x²+y²-2x-4y+3=0, we need to find the point of contact by solving the system of equations formed by the line and the circle.
Step 1: Rewrite the equation of the line x+y=5 in the form y = mx + c, where m is the slope and c is the y-intercept:
y = -x + 5
Step 2: Substitute the expression for y into the equation of the circle:
x² + (-x + 5)² - 2x - 4(-x + 5) + 3 = 0
Step 3: Simplify the equation:
x² + (x² - 10x + 25) - 2x + 4x - 20 + 3 = 0
2x² - 8x + 8 = 0
Step 4: Solve the quadratic equation:
Divide both sides of the equation by 2 to simplify:
x² - 4x + 4 = 0
(x - 2)(x - 2) = 0
(x - 2)² = 0
Step 5: Find the value of x:
x - 2 = 0
x = 2
Step 6: Substitute the value of x into the equation of the line to find y:
y = -x + 5
y = -2 + 5
y = 3
Therefore, the point of contact between the line x+y=5 and the circle x²+y²-2x-4y+3=0 is (2, 3).
To show that the line x+y=5 touches the circle x²+y²-2x-4y+3=0, we need to find the point of contact between the line and the circle.
Step 1: Convert the equation of the line into slope-intercept form (y = mx + c).
x + y = 5
y = -x + 5
Step 2: Substitute the value of y from the line equation into the equation of the circle to get a quadratic equation in terms of x.
x² + (-x + 5)² - 2x - 4(-x + 5) + 3 = 0
Simplifying the equation:
x² + (x² - 10x + 25) - 2x + 4x - 20 + 3 = 0
2x² - 8x + 8 = 0
Step 3: Determine the discriminant of the quadratic equation to check if it has real solutions.
The discriminant (D) is calculated as: D = b² - 4ac
In the quadratic equation, ax² + bx + c = 0, a = 2, b = -8, c = 8
D = (-8)² - 4(2)(8)
D = 64 - 64
D = 0
Since the discriminant is zero, this implies that the quadratic equation has a single real root.
Step 4: Find the value of x using the quadratic formula.
The quadratic formula is given by: x = (-b ± √D) / (2a)
Since D = 0, we can simplify the formula to: x = -b / (2a)
x = -(-8) / (2 * 2)
x = 8 / 4
x = 2
Step 5: Substitute the value of x into the equation of the line to find the corresponding value of y.
y = -x + 5
y = -2 + 5
y = 3
Therefore, the point of contact between the line x + y = 5 and the circle x² + y² - 2x - 4y + 3 = 0 is (2, 3).