TS EAMCET · Maths · Circle

If $9 x^{2} - 24 x y + 16 y^{2} + α x + β y + 6 = 0$ represents a pair of parallel lines of $1$ unit apart and one of those lines passes through $(1, 1)$ then $\frac{α}{β} =$

A $\frac{2}{3}$
B $1$
C $\frac{- 3}{2}$
D $\frac{- 3}{4}$

Verified Solution

Answer & Solution

Correct Answer

(D) $\frac{- 3}{4}$

Step-by-step Solution

Detailed explanation

If ax2+2hxy+by2+2gx+2fy+c=0 represents a pair of parallel lines then condition applies h2=ab & bg2=af2 Compare the given pair of lines with ax2+2hxy+by2+2gx+2fy+c=0 ,we get a=9,2h=-24,b=16,α=2g,β=2f,c=6 Put all given values in bg2=af2 we get…

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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Step-by-step Solution

If $9 x^{2} - 24 x y + 16 y^{2} + α x + β y + 6 = 0$ represents a pair of parallel lines of $1$ unit apart and one of those lines passes through $(1, 1)$ then $\frac{α}{β} =$