Assertion :Let AB & CD be parallel chords of the circle |z| = 1, Let point A, B, C, D represented by complex number $z_{1}, z_{2}, z_{3}, z_{4}$ respectively.
The distance between the chords AB & CD is $\frac{1}{2} | z_{1} - z_{3} | | z_{2} - z_{3} |$ . Reason: The Area of the triangle $A B C = \frac{a b c}{4}$ .

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 A Both (A) & (R) are individually true & (R) is correct explanation of (A).
Area of $Δ A B C$ is $A B = | z_{1} - z_{2} | | z_{2} - z_{3} | | z_{3} - z_{1} |$
$∴$ Altitude of $Δ A B C = \frac{2 A r e a o f Δ A B C}{b a s e}$
$= \frac{2 | z_{1} - z_{2} | | z_{2} - z_{3} | | z_{3} - z_{1} |}{4 x | z_{1} - z_{2} |}$
$= \frac{| z_{2} - z_{3} | | z_{3} - z_{1} |}{2}$
$= \frac{1}{2} | z_{2} - z_{3} | | z_{3} - z_{1} |$
Hence Assertion (A) & Reason (R) both are true & Reason (R) is the correct explanation of Assertion (A).

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Similar questions

State True=1 and False=0

z_{1}, z_{2}, z_{3}

are three distinct complex numbers and p, q, r are three positive real numbers such that

\frac{p}{| z_{2} - z_{3} |} = \frac{q}{| z_{3} - z_{1} |} = \frac{r}{| z_{1} - z_{2} |}

then

\frac{p^{2}}{z_{2} - z_{3}} + \frac{q^{2}}{z_{3} - z_{1}} + \frac{r^{2}}{z_{1} - z_{2}} = 0

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 (A)
If the circumferences of two circles are in the ratio 2 : 3, then the ratio of their areas is 4 : 9.

Reason (R)
The circumference of a circle of radius r is 2πr.

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) Reason (R) 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.

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]

Q. Assertion (A)
In ∆ABC, if DE ∥ BC intersects AB in D and AC in E, then

\frac{A D}{A B} = \frac{A E}{A C} .

Reason (R)
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then these sides are divided in the same ratio.

(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 AB & CD be parallel chords of the circle |z| = 1, Let point A, B, C, D represented by complex number z1,z2,z3,z4 respectively.The distance between the chords AB & CD is 12|z1−z3||z2−z3|. Reason: The Area of the triangle ABC=abc4.