Question: Find the value of p, if the lines represented by the following equations coincide to each other: $2x² +pxy +2y² = 0$.

Solution:

Given,
Equation of line is $2x² +pxy +2y² = 0$
$or, 2x² +2×\frac{p}{2}xy + 2y² = 0$ - (i)

Comparing equation (i) with $ax² + 2hxy + by²=0$, we get,
$a = 2, h = \frac{p}{2}, and b = 2$

We know, when two lines are coincident, $h² = ab$,
$or, h² = ab$
$or, 2² = \frac{p}{2} × 2$
$or, 4 = p$
$\therefore p = 4$

Therefore, the required value of p, when the lines represented by the given equation coincide to each other is 4.

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