Answer:
a) 5
Step-by-step explanation:
2x² - 6x + 2y² + 2y = 45
Complete the square and get it in the form (x-a)² + (y -b)² = r²
2x² - 6x + 2y² + 2y = 45
Divide the equation by 2
x² - 3x + y² + y = 45/2
x² - 3x + ? + y² + y + ?? = 45/2
? -----> (x coefficient by 2)² = (3/2)² = 9/4
?? ----> (y coefficient by 2)² = (1/2)² = 1/4
Add (9/4) & (1/4) to both sides,
<u>x² -3x +(9/4)</u> + <u>y² + y + (1/4)</u> = (45/2)+ (9/4) + (1/4)
(x - 3/2)² + (y + 1/2)² = (90/4) + (9/4) + (1/4)
Radius = 5
The value of x in the triangle is (d) 7
<h3>How to solve for x?</h3>The complete question is in the attached image.
From the attached image of the triangle, we can see that the triangle is a right triangle, and x can be solved using the following sine function
Evaluate sin(45)
Solve for x
Divide
x = 7
Hence, the value of x in the triangle is (d) 7
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The area of the triangle with the vertices (0, 5), (2, -2), and (5, 1) by using the calculus is 21 square unit.
We need to find the equation among all possible pairs and then integrate the equations from one co-ordinate to another co-ordinate
Equation of line passing through (0,5) and (2,-2) is
y-5 = [(-2-5)/(2-0)](x-0)
=>y-5 = (-7) /2x
=>y= (-7/2x)+5 -------(eq1)
Equation of line passing through (0,5) and (5,1) is
y-5 = [(1-5) / (5-0)](x-0)
=>y-5 = (-4/5)x
=>y = (-4/5)x+5-------(eq2)
Equation of line passing from (2,-2) and (5,1) is
y-(-2) = [[1 - (-2)] / (5-2)] / (x-2)
=>y+2=(3/3)(x-2)
=>y=x-4--------(eq3)
Now, we use definite integration to find the area between the different equation of line.
So, area enclosed between the equations is given by the
area =[(-4/5)x+5 - (-7/2)x + 5)dx +
[(-4/5)x+5 -(x-4)]dx
=>area=(7/2-4/5)x dx +
((-9/5)x+9)dx
Using properties of integration,
=>area=(27/10)x dx +
(-9/5)x+9)dx
=>area=([27/10)×[5² - 2²])/2 + [ (-9/5)×(5²-1²) ]/2 +9×(5-1)
=>area=(27/20)×(25-4) + (-9/5)×24+9×4
=>area = (27×21)/20 + (-216)/5+ 36
=>area=(567/20) - (216/5) + 36
=>area= [(567-261×4)+(36×20)]/20
=>area=[(567-864)+720]/20
=>area=423/20
=>area=21 square unit.
Hence, area of triangle is 21 square unit.
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