Locus and Complex Expressions

The video tutorial explores the conditions under which the expression z minus 1 on z minus i is real or imaginary. It discusses the importance of the numerator and denominator in determining the real and imaginary parts, highlighting the discontinuity at z equals i. The tutorial also covers completing the square to identify the shape of the locus and emphasizes the need to exclude certain values to avoid mathematical inconsistencies. The conclusion summarizes the key points and encourages careful diagramming for geometric reasoning.

What condition must be met for the expression (z-1)/(z-i) to be real?

• The real part must be zero.

• The imaginary part must be zero.

• Both real and imaginary parts must be zero.

• The expression must be undefined.

Why is the point z = i excluded from the locus?

• It is not on the line.

• It results in a complex number.

• It is a point of intersection.

• It makes the expression undefined.

What is the gradient of the line derived from the expression (z-1)/(z-i) being real?

• 1

• 0

• 2

• -1

What is the y-intercept of the line derived from the expression (z-1)/(z-i) being real?

• 0

• 1

• -1

• 2

What additional point must be excluded when considering the expression (z-1)/(z-i) to be imaginary?

• z = 1

• z = 2

• z = -1

• z = 0

What happens if both the real and imaginary parts of the expression are zero?

• The expression is zero.

• The expression is an imaginary number.

• The expression is a real number.

• The expression is undefined.

What shape is formed by the locus when the expression (z-1)/(z-i) is imaginary?

• An ellipse

• A circle

• A parabola

• A straight line

What is the center of the circle formed by the locus?

• (1/2, 1/2)

• (1, 1)

• (1/2, 1/2i)

• (0, 0)

What is the radius of the circle formed by the locus?

• 1/√2

• 1

• √2

• 1/2

What mathematical concept is used to determine the shape of the locus?

• Completing the square

• Differentiation

• Matrix multiplication

• Integration