Final answer:
(i) To express the equation y = x² – 4x + 7 in the form (x – a)² + b, we need to complete the square. The equation can be written as (x – 2)² + 3, and the coordinates of the minimum point on the curve are (2, 3). (ii) The value of m for which the line y = mx – 3m is a tangent to the curve y = x² – 4x + 7 is 1. However, there are no points where the line touches the curve for m = 1.
Step-by-step explanation:
(i) To express the equation y = x² – 4x + 7 in the form (x – a)² + b, we need to complete the square. First, let's rearrange the equation:
y = (x² – 4x) + 7
Now, let's complete the square for the quadratic term (x² – 4x):
y = (x² – 4x + 4) + 7 – 4
Next, simplify the equation:
y = (x – 2)² + 3
Therefore, the equation y = x² – 4x + 7 can be written in the form (x – 2)² + 3. The coordinates of the minimum point on the curve are (2, 3).
(ii) To find the values of m for which the line y = mx – 3m is a tangent to the curve y = x² – 4x + 7, we need to equate the equations and solve for x. Setting the equations equal to each other:
(x – 2)² + 3 = mx – 3m
Simplifying the equation:
x² – 4x + 4 + 3 = mx – 3m
Combining like terms:
x² – 4x + 7 = mx – 3m
From here, we can equate the coefficients of x on both sides:
1 = m
Therefore, the value of m for which the line is a tangent to the curve is 1. Substituting this value into the equation y = mx – 3m, we can find the coordinates of the point where the line touches the curve. For m = 1:
y = x – 3
Setting the equation equal to y = x² – 4x + 7:
x² – 4x + 7 = x – 3
Simplifying the equation:
x² – 5x + 10 = 0
Using the quadratic formula to solve for x:
x = (-(-5) ± √((-5)² – 4(1)(10))) / 2(1)
Simplifying the expression:
x = (5 ± √(25 – 40)) / 2
Since the discriminant is negative, there are no real solutions for x. Therefore, there are no points where the line touches the curve for m = 1.