TS EAMCET · Maths · Pair of Lines

Let $L_{1}, L_{2}$ be the lines represented by the equation $4 x^{2} - 5 x y + 3 y^{2} = 0$ . Let $L_{3}, L_{4}$ be two lines passing through the point $(4, 3)$ such that $L_{3}$ and $L_{4}$ are perpendicular to $L_{1}$ and $L_{2}$ respectively. If the combined equation of $L_{3}$ and $L_{4}$ is $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$ , then $a f + b g + c h =$

A $144$
B $66$
C $78$
D $216$

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Answer & Solution

Correct Answer

(D) $216$

Step-by-step Solution

Detailed explanation

Let L1, L2 be the lines represented by the equation 4x2-5xy+3y2=0, Now comparing with ax2+2hxy+by2=0 we get, m1+m2=-2hb=53 and m1m2=ab=43 Now let slope of perpendicular line L3, L4 be -1m1 & -1m2 which is passing through the point 4,3 So, equation of pair…

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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Let L1, L2 be the lines represented by the equation 4x2-5xy+3y2=0. Let L3, L4 be two lines passing through the point 4,3 such that L3 and L4 are perpendicular to L1 and L2 respectively. If the combined equation of L3 and L4 is ax2+2hxy+by2+2gx+2fy+c=0, then af+bg+ch=

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