MHT CET · Maths · Pair of Lines

If the equation \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y=0\) has one line as the bisector of the angle between co-ordinate axes, then

A \((a+b)^{2}=4\left(h^{2}+g^{2}\right)\)
B \((a+b)^{2}=4 h^{2}\)
C \((a+b)^{2}=4\left(h^{2}+f^{2}\right)\)
D \((a+b)^{2}=4\left(h^{2}+g^{2}+f^{2}\right)\)

Verified Solution

Answer & Solution

Correct Answer

(B) \((a+b)^{2}=4 h^{2}\)

Step-by-step Solution

Detailed explanation

(A)
In the given pair of lines, one line is \(y=\pm x \Rightarrow x \pm y=0\)
Let the other line be \(a x+b y+c=0\)
\(\therefore(a x+b y+c)(x+y)=0\) or \((a x+b y+c)(x-y)=0\)
\(a x^{2}+b x y+c x+a x y+b y^{2}+c y=0\) or \(a x^{2}+b x y+c x-a x y-b y^{2}-c y=0\)
\(a x^{2}+(a+b) x y+b y^{2}+c x+c y=0 \quad \ldots(1)\) or \(a x^{2}+(b-a) x y-b y^{2}+c x-c y=0 \ldots(2)\)
Given eq. is \(a x^{2}+2 h x y+b y^{2}+2 g x+2 f y=0 \quad \ldots\) (3)
Eq. (1) and (3) as well as eq. (2) and (3) represent the same line.
Comparing, we write
\(2 h=a+b\) or \(2 h=b-a \Rightarrow 4 h^{2}=(a+b)^{2}\) among options given.

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.

Same subject

Explore more questions on app

From MHT CET

Explore more questions on app