If α,β are the roots of ax2+bx+c=0;α+h,β+h are the roots of px2+qx+r=0; and D1,D2 the respective discriminants of these equation then D1:D2=

A
$\frac{a^{2}}{p^{2}}$
B
$\frac{b^{2}}{q^{2}}$
C
$\frac{c^{2}}{r^{2}}$
D
None of these

Solution:

Let, $A = α + h and B = β + h$ , then

$\begin{aligned} A - B = (α + h) - (β + h) = α - β \\ \Rightarrow & (A - B)^{2} = (α - β)^{2} \\ \Rightarrow & (A + B)^{2} - 4 AB = (α + β)^{2} - 4 αβ \\ \Rightarrow & \frac{q^{2}}{p^{2}} - \frac{4 r}{p} = \frac{b^{2}}{a^{2}} - \frac{4 c}{a} \\ \Rightarrow & \frac{b^{2} - 4 ac}{a^{2}} = \frac{q^{2} - 4 p r}{p^{2}} \Rightarrow \frac{D_{1}}{a^{2}} = \frac{D_{2}}{p^{2}} \Rightarrow \frac{D_{1}}{D_{2}} = \frac{a^{2}}{p^{2}} \end{aligned}$

If $α, β$ are the roots of ${ax}^{2} + bx + c = 0; α + h, β + h$ are the roots of ${px}^{2} + qx + r = 0; and D_{1}, D_{2}$ the respective discriminants of these equation then $D_{1} : D_{2} =$

Solution:

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