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Byju's Answer

Standard XII

Mathematics

Distance between Parallel Lines

If the system...

Question

If the system of equations $2 x + 3 y = 7$ and $2 a x + (a + b) y = 28$ represents coincident lines, which of the conditions holds true?

b = 2 a

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a = 2 b

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2 a + b = 0

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a + 2 b = 0

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Solution

The correct option is B $b = 2 a$
$2 x + 3 y = 7$
$2 a x + (a + b) y = 28$
For incident lines, condition is
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\Rightarrow \frac{2}{2 a} = \frac{3}{a + b} = \frac{7}{28}$
$\Rightarrow \frac{1}{a} = \frac{3}{a + b}$
$\Rightarrow a + b = 3 a$
$\Rightarrow b = 2 a$
Hence, the correct option is (A).

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Similar questions

Q. Assertion :If the system of equations

2 x + 3 y = 7

and

2 a x + (a + b) y = 28

has infinitely many solutions, then

2 a - b = 0

Reason: The system of equations

3 x - 5 y = 9

and

6 x - 10 y = 8

has a unique solution.

Q. If the system of equations has infinitely many solutions, then

2 x + 3 y = 7 2 a x + (a + b) y = 28

(a) a = 2b
(b) b = 2a
(c) a + 2b = 0
(d) 2a + b = 0

Determine the nature of the roots of the following quadratic equations;
$(i) (x - 2 a) (x - 2 b) = 4 a b (i i) 9 a^{2} b^{2} x^{2} - 24 a b c d x + 16 c^{2} d^{2} = 0, a \neq 0, b \neq 0 (i i i) 2 (a^{2} + b^{2}) x^{2} + 2 (a + b) x + 1 = 0 (i v) (b + c) x^{2} - (a + b + c) x + a = 0$

Find the values of a and b for which the following system of equations has infinitely many solutions:

(i) (2a - 1) x - 3y = 5

3x + (b - 2) y = 3

(ii) 2x - (2a + 5) y = 5

(2b + 1) x - 9y = 15

(iii) (a - 1) x + 3y = 2

6x + (1 - 2b) y = 6

(iv) 3x + 4y = 12

(a + b) x + 2 (a - b) y = 5a - 1

(v) 2x + 3y = 7

(a - b) x + (a + b) y = 3a + b - 2

(vi) 2x + 3y - 7 = 0

(a - 1) x + (a + 1) y = (3a - 1)

(vii) 2x + 3y = 7

(a - 1) x + (a + 2) y = 3a

(viii) x + 2y = 1

(a - b) x + (a + b) y = a + b - 2

(ix) 2x + 3y = 7

2ax + ay = 28 - by

Q. Solve the following systems of equations by the method of cross-multiplication:

2 a x + 3 b y = a + 2 b

3 a x + 2 b y = 2 a + b

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If the system of equations 2x+3y=7 and 2ax+(a+b)y=28 represents coincident lines, which of the conditions holds true?

The correct option is B b=2a2x+3y=72ax+(a+b)y=28For incident lines, condition isa1a2=b1b2=c1c2⇒22a=3a+b=728⇒1a=3a+b⇒a+b=3a⇒b=2aHence, the correct option is (A).

If the system of equations $2 x + 3 y = 7$ and $2 a x + (a + b) y = 28$ represents coincident lines, which of the conditions holds true?