Given $x^{2}+y^{2}=R^{2}$, so that we multiply every $x$ by $a$ and every $y$ by $b$, $(a>b)$
And the distance between the focuses of this locus is $48R$, and the area of the rhombus which Vertices are on the $x$-axis and $y$-axis ,bounded in that locus, is $350$.
Need to find $a,b$.
So i find that the locus is $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=R^{2}$ and tried to calculate by distance formula but got a very messy calculation.
Would be glad to see any ideas.
Thanks!
Your first statement does not lead to the equation for the ellipse that you give later, so I will pass over that and simply use the standard-form equation, $$\frac{x^2}{(aR)^2} + \frac{y^2}{(bR)^2} = 1 .$$
This tells us that the ellipse is centered on the origin with semi-horizontal axis $aR$ and semi-vertical axis $bR$; the requirement that $ a > b $ makes the result unique because the major (focal) axis of the ellipse must then be "horizontal" (not having this inequality would also permit a "vertical" ellipse of the same size).
The area of the rhombus is four times the area of a triangle with its legs being the semi-horizontal and semi-vertical axes, so we have $$4 \cdot \frac{1}{2} \cdot aR \cdot bR = 2abR^2 = 350 \Rightarrow abR^2 = 175 .$$
The focal distance $c$ of a focus from the center of the ellipse is given by $c^2 = a^2 - b^2$ . The separation of the foci is then $2c = 48R$ , so we may write $$4c^2 = 4(a^2 - b^2) = 48^2R^2 \Rightarrow a^2 - b^2 = 12 \cdot 48 \ R^2 $$
$$\Rightarrow a^2 - (\frac{175}{aR^2})^2 = 12 \cdot 48 \ R^2 . $$
Multiplying through by $a^2$ will give us a quadratic equation in $a^2$:
$$a^4 - (12 \cdot 48 \ R^2) \cdot a^2 - (\frac{175^2}{R^4}) = 0 $$
$$ \Rightarrow a^2 = \frac{(12 \cdot 48 \ R^2) + \sqrt{(12 \cdot 48 \ R^2)^2 + 4 \cdot (\frac{175^2}{R^4})}}{2} = (288) \cdot \left[1 + \sqrt{1 + (\frac{175}{288 \cdot R^4})^2} \right] \cdot R^2 ,$$
where we have discarded the "negative radical" solution from the quadratic formula, since the radical is larger than $12 \cdot 48 R^2$ alone. We can then insert this into our earlier relation $b^2 = (\frac{175}{aR^2})^2 $.
Note that we are not given enough information to find specific values for all three variables; the proportions of the ellipse are determined, but the size of the ellipse is determined by the choice for $R$ .