AP EAMCET · Maths · Straight Lines

For three consecutive odd integers $a, b$ and $c$ , if the variable line $a x + b y + c = 0$ always passes through the point $(α, β)$ , the value of $α^{2} + β^{2}$ equals

A $9$
B $4$
C $5$
D $3$

Verified Solution

Answer & Solution

Correct Answer

(C) $5$

Step-by-step Solution

Detailed explanation

The given line is ax+by+c=0. Since a, b and c are three consecutive odd integers, so they are in arithmetic progression, ⇒b-a=c-b ⇒c=2b-a Hence, the equation of line is ax+by+2b-a=0 ⇒ax-1+by+2=0. Now, the is given that the line always pass through point…

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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For three consecutive odd integers a, b and c, if the variable line ax+by+c=0 always passes through the point (α,β), the value of α2+β2 equals

Step-by-step Solution

For three consecutive odd integers $a, b$ and $c$ , if the variable line $a x + b y + c = 0$ always passes through the point $(α, β)$ , the value of $α^{2} + β^{2}$ equals