Question: Find the value of k, if the lines represented by the following equations coincide to each other: $(k-9)x² -(2k-10)xy +ky² = 0$.

Solution:

Given,
Equation of line is $(k-9)x² -(2k-10)xy +ky² = 0$
$or, (k-9)x² - 2(k-5)xy + ky² = 0$ - (i)

Comparing equation (i) with $ax² + 2hxy + by²=0$, we get,k
$a = (k-9), h = -(k-5), and b = k$

We know, when two lines are coincident, $h² = ab$,
$or, h² = ab$
$or, \{-(k-5)\}² = (k-9)k$
$or, \{5 -k\}² = (k-9)k$
$or, 25 - 10k +k² = k² -9k$
$or, 25 -10k +9k +k² -k² = 0$
$or, 25 - k = 0$
$\therefore k = 25$

Therefore, the required value of p, when the lines represented by the given equation coincide to each other is 25.

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