Question

1.      a2
– b 2– c2 – 2bc = 0 , then the family of lines ax + by +
c = 0 are concurrent at the points:

  a. (0,1)  b. (1,-1) c. (-1,1) d. none of the above

Answer 1

i think the answer can be 0 or 1

Answer 2

Answer:

(c)The given family of lines is concurrent at the point (-1,1)

Step-by-step explanation:

The given algebraic relation is

$a {}^{2} - b {}^{2} - c {}^{2} - 2bc = 0$

And the given family of lines is

$ax + by + c = 0 - - (1)$

The algebraic relation can be rearranged as follows

$a {}^{2} - b {}^{2} - c {}^{2} - 2bc = 0 \\ = > a {}^{2} = b {}^{2} + c {}^{2} + 2(b)(c) \\ = > a {}^{2} =( b + c) {}^{2} \\ = > a = b + c \\ = > c = a - b - - > (2)$

Substituting the equation (2) in equation (1),we get

$ax + by + (a - b) = 0 \\ = > a(x + 1) + b(y - 1) = 0$

The above relation is only possible when both the x and y terms are zero.Therefore

$x + 1 = 0 = > x = - 1 \\ y - 1 = 0 = > y = 1$

Therefore,the given family of lines is concurrent at the point (-1,1)

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