AP EAMCET · Maths · Straight Lines

If the lines represented by $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ intersect on the $x$ -axis, which of the following is in general incorrect

A $a b c = 2 f g h$
B $g^{2} = a c$
C $a f^{2} = c h^{2}$
D $a f^{2} + c h^{2} = 2 f g h$

Verified Solution

Answer & Solution

Correct Answer

(A) $a b c = 2 f g h$

Step-by-step Solution

Detailed explanation

Given, ax2+2hxy+by2+2gx+2fy+c=0 put y=0 ⇒ax2+2gx+c=0 Now roots will be equal to given axis so, D=0⇒g2=ac ......1 Now for pair of straight line abc+2fgh-af2-bg2-ch2=0 Since g2=ac So 2fgh=af2+ch2 .......2 Also 2fgh2=af2+ch22 ⇒4f2g2h2=af2+ch22…

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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If the lines represented by ax2+2hxy+by2+2gx+2fy+c=0 intersect on the x-axis, which of the following is in general incorrect

Step-by-step Solution

If the lines represented by $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ intersect on the $x$ -axis, which of the following is in general incorrect