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Standard XII

Mathematics

Addition of Vectors

Assertion :Le...

Question

Assertion :Let $z$ be a complex number.The equation $| z - 1 | + | z - i | = 2$ represents an ellipse. Reason: The equation $| z - z_{1} | + | z - z_{2} | = k$ where k is positive, represents an ellipse.

Both (A) & (R) are individually true & (R) is correct explanation of (A).

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Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).

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(A) is true (R) is false.

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(A) is false (R) is true.

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Solution

The correct option is C (A) is true (R) is false.
Reason is true only if $k > | z_{1} - z_{2} |$
Here $k = 2, z_{1} = 1, z_{2} = i$
and $| z_{1} - z_{2} | = | 1 - i | = \sqrt{2} \Rightarrow 2 > | 1 - i | = \sqrt{2}$
$\Rightarrow$ Reason is false but Assertion is true.

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Similar questions

Q. Assertion :Let z be a complex number.
The equation |z - 1| - |z - i| = 1 represents a part of hyperbola. Reason: The equation

| z - z_{1} | - | z - z_{2} | = k

is a hyperbola.

Q. Assertion (A): At constant temperature,

P V

P

plot for real gases is not a straight line.
Reason (R): At high pressure all gases have

(Z > 1)

but at intermediate pressure most gases have

(Z < 1)

DIRECTIONS (Qs. 52 AND 53): The following item consist of two statements: one labelled as the Assertion a) and the other as Reason (R). You are to examine these two statements carefully and select the answers to these items using the codes given below:

a) Both A and R are individually true but R is the correct explanation of A

b) Both A and R are individually true but R is not the correct explanation of A

c) A is true but R is false

d) Both A and R are false

53. Assertion a): Saluva Narasimha put an end to the old dynasty and assumed the royal title.

Reason

(R): He wanted to save the kingdom from further degeneration and disintegration. [2003]

In the following question, a statement of Assertion (A) followed by a statement of Reason (R) is given. Choose the correct option out of the choices given below in each question.

Assertion (A) At constant temperature, PV vs V plot for real gases is not a straight line.

Reason (R) At high pressure all gases have Z > 1 but at an intermediate pressure, most gases have Z < 1.

(a) Both A and R are true and R is the correct explanation of A

(b) Both A and R are true R is not the correct explanation of A

(d) A is false but R is true

a) Both A and R are individually true but R is the correct explanation of A

b) Both A and R are individually true but R is not the correct explanation of A

c) A is true but R is false

d) Both A and R are false

52. Assertion a): Shah Alam II spent the initial years as an emperor far away from his capital.

Reason

(R): There was always a lurking danger of foreign invasion from the north-west frontier. [2003]

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Assertion :Let z be a complex number.The equation |z−1|+|z−i|=2 represents an ellipse. Reason: The equation |z−z1|+|z−z2|=k where k is positive, represents an ellipse.

The correct option is C (A) is true (R) is false.Reason is true only if k>|z1−z2|Here k=2,z1=1,z2=iand |z1−z2|=|1−i|=√2⇒2>|1−i|=√2⇒ Reason is false but Assertion is true.

Assertion :Let $z$ be a complex number.The equation $| z - 1 | + | z - i | = 2$ represents an ellipse. Reason: The equation $| z - z_{1} | + | z - z_{2} | = k$ where k is positive, represents an ellipse.

The correct option is C (A) is true (R) is false.
Reason is true only if $k > | z_{1} - z_{2} |$
Here $k = 2, z_{1} = 1, z_{2} = i$
and $| z_{1} - z_{2} | = | 1 - i | = \sqrt{2} \Rightarrow 2 > | 1 - i | = \sqrt{2}$
$\Rightarrow$ Reason is false but Assertion is true.