TS EAMCET · Maths · Pair of Lines

A pair of straight lines is passing through the point $(1, 1)$ . One of the lines makes an angle $θ$ with the positive direction of $X$ -axis and the other makes the same angle with the positive direction of $Y$ -axis. If the equation of the pair of straight lines is $x^{2} - (a + 2) x y + y^{2} + a (x + y - 1) = 0, a \neq - 2$ , then the value of $θ$ is

A $\frac{1}{2} \sin^{- 1} (\frac{2}{a + 2})$
B $\frac{1}{2} \sin (\frac{2}{a + 2})$
C $\frac{1}{2} \tan^{- 1} (\frac{2}{a + 2})$
D $\frac{1}{2} \tan (\frac{2}{a + 2})$

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{1}{2} \sin^{- 1} (\frac{2}{a + 2})$

Step-by-step Solution

Detailed explanation

Given pair of straight lines is, x2-a-2xy+y2+ax+y-1=0 ...1 and the point is (1,1). The slopes of straight lines are m1=tanθ; m2=tan90-θ m2=cotθ The equations of straight lines are y-1=tanθx-1 ...2 and…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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