6these r actually some general derivations so it wud be better if u see them properly!!
1.find the condition that the expression ax2+2hxy+by2 may have two factors of the form (y-mx) and (my+x) respectively.
ans: a+b=0
2.find the condition that ax2+2hxy+by2 and a1x2+2h1xy+b1y2 may have factors of the form (y-mx) and (my+x) respectively.
ans:(aa1-bb1)2+4(bh1+a1h)(b1h+ah1)=0
3.if x is real,prove that the expression (x-a)(x-c)/(x-b) is capable of assuming all values if a,b,c are either in increasing or in decreasing order of magnitude.
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3 Answers
3411) If you let t = \frac{y}{x}, then we can reformulate the problem as, under what conditions will the roots of the quadratic, at^2+2ht+b=0 have roots, m and -1/m.
Since product of roots is -1, we must have \frac{b}{a}=-1\Rightarrow a+b=0
2) Now, you can try this one
3) WLOG a<b<cf = \frac{(x-a)(x-c)}{(x-b)}; f: (-\infty, b) \rightarrow \mathbb{R}
It is continuous everywhere in its domain.
Now, as x \rightarrow -\infty , f(x) \rightarrow -\infty and asx \rightarrow b , f(x) \rightarrow \infty
Since its continuous, from Mean Value Theorem f takes all real values.
A similar analysis for (b, \infty) shows us that the given expression takes on every real value at least twice
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