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Byju's Answer

Standard XII

Mathematics

Area between x=g(y) and y Axis

Assertion :Le...

Question

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.

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.
The equation $| | z - z_{1} | - | z - z_{2} | | = k$ is a hyperbola only if
$k < | z_{1} - z_{2} |$
Here $k = 1$ & $| z_{1} - z_{2} | = \sqrt{2} ≮ 1$
$z_{1} = 1, z_{2} = i$
$\Rightarrow$ Reason is false and Assertion is true.

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Similar questions

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)

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

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]

Q. 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.

Q. Assertion (A)
The equation x² + x + 1 = 0 has both real roots.

Reason (R)
The equation ax2 + bx + c = 0, (a ≠ 0) has both real roots, if (b2 − 4ac) > 0.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

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Assertion :Let z be a complex number.The equation |z - 1| - |z - i| = 1 represents a part of hyperbola. Reason: The equation |z−z1|−|z−z2|=k is a hyperbola.

The correct option is C (A) is true (R) is false.The equation ||z−z1|−|z−z2||=k is a hyperbola only ifk<|z1−z2|Here k=1 & |z1−z2|=√2≮1z1=1,z2=i⇒ Reason is false and Assertion is true.

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.

The correct option is C (A) is true (R) is false.
The equation $| | z - z_{1} | - | z - z_{2} | | = k$ is a hyperbola only if
$k < | z_{1} - z_{2} |$
Here $k = 1$ & $| z_{1} - z_{2} | = \sqrt{2} ≮ 1$
$z_{1} = 1, z_{2} = i$
$\Rightarrow$ Reason is false and Assertion is true.