MHT CET · Maths · Pair of Lines

If sum of the slopes of the lines given by $x^{2} - 4 p x y + 8 y^{2} = 0$ is three times their product then $p =$ …

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

Verified Solution

Answer & Solution

Correct Answer

(A) $\frac{3}{4}$

Step-by-step Solution

Detailed explanation

Given, pair of lines,

x^{2} - 4 p x y + 8 y^{2} = 0

Which is in form of

a x^{2} + 2 h x y + b y^{2} = 0

According to question,
Sum of the slopes

= 3 \times

product of the slopes

\Rightarrow m_{1} + m_{2} = 3 \times (m_{1} m_{2})

\Rightarrow \frac{- (- 4 p)}{8} = 3 \times \frac{1}{8}

\Rightarrow 4 p = 3 \Rightarrow p = \frac{3}{4}

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 sum of the slopes of the lines given by $x^{2} - 4 p x y + 8 y^{2} = 0$ is three times their product then $p =$ …