Question
Solve for x and y:
$a^2x + b^2y = c^2,$
$b^2x + a^2y = d^2$
$a^2x + b^2y = c^2,$
$b^2x + a^2y = d^2$
✓
Answer
$a^2x + b^2y = c^2...(i)$
$b^2x + a^2y = d^2...(ii)$
Multiplying (i) by $a ^2$ and (ii) by $b ^2$ and subtracting, we get
$\Rightarrow\text{a}^4\text{x} - \text{b}^4\text{x} = \text{a}^2\text{b}^2 - \text{b}^2\text{d}^2$
$\Rightarrow\text{x}=\frac{\text{a}^2\text{c}^2-\text{b}^2\text{d}^2}{\text{a}^4-\text{b}^4}$
Multiplying (i) by $a ^2$ and (ii) by $b ^2$ and subtracting, we get
$\Rightarrow\text{b}^4\text{y} - \text{a}^4\text{y} = \text{b}^2\text{c}^2 - \text{a}^2\text{d}^2$
$\Rightarrow\text{y}=\frac{\text{b}^2\text{c}^2-\text{a}^2\text{d}^2}{\text{b}^4-\text{a}^4}$
So, $\text{x}=\frac{\text{a}^2\text{c}^2-\text{b}^2\text{d}^2}{\text{a}^4-\text{b}^4}$ and $\text{y}=\frac{\text{b}^2\text{c}^2-\text{a}^2\text{d}^2}{\text{b}^4-\text{a}^4}$
$b^2x + a^2y = d^2...(ii)$
Multiplying (i) by $a ^2$ and (ii) by $b ^2$ and subtracting, we get
$\Rightarrow\text{a}^4\text{x} - \text{b}^4\text{x} = \text{a}^2\text{b}^2 - \text{b}^2\text{d}^2$
$\Rightarrow\text{x}=\frac{\text{a}^2\text{c}^2-\text{b}^2\text{d}^2}{\text{a}^4-\text{b}^4}$
Multiplying (i) by $a ^2$ and (ii) by $b ^2$ and subtracting, we get
$\Rightarrow\text{b}^4\text{y} - \text{a}^4\text{y} = \text{b}^2\text{c}^2 - \text{a}^2\text{d}^2$
$\Rightarrow\text{y}=\frac{\text{b}^2\text{c}^2-\text{a}^2\text{d}^2}{\text{b}^4-\text{a}^4}$
So, $\text{x}=\frac{\text{a}^2\text{c}^2-\text{b}^2\text{d}^2}{\text{a}^4-\text{b}^4}$ and $\text{y}=\frac{\text{b}^2\text{c}^2-\text{a}^2\text{d}^2}{\text{b}^4-\text{a}^4}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If the angle of elevation of a cloud from a point h metres above a lake is a and the angle of depression of its reflection in the lake be b, prove that the distance of the cloud from the point of observation is, $\frac{2\text{h}\sec\alpha}{\tan\beta-\tan\alpha}.$
→Solve the following system of equations by the method of cross-multiplication:
$\frac{\text{x}}{\text{b}}=\frac{\text{y}}{\text{b}},$
$\text{ax}+\text{by}=\text{a}^2+\text{b}^2.$
→$\frac{\text{x}}{\text{b}}=\frac{\text{y}}{\text{b}},$
$\text{ax}+\text{by}=\text{a}^2+\text{b}^2.$
Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1) , (1, 3) and (x, 8) respectively.
200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on (see Fig.). In how many rows are the 200 logs placed and how many logs are in the top row?
Solve the following pairs of equations:
$\text{x}+\text{y}=3.3,\ \frac{0.6}{3\text{x}-2\text{y}}=-1,\ 3\text{x}-2\text{y}\neq0$
→$\text{x}+\text{y}=3.3,\ \frac{0.6}{3\text{x}-2\text{y}}=-1,\ 3\text{x}-2\text{y}\neq0$
Find the area of the flower bed $($with semi$-$circular ends$)$ shown in Fig.
→Abdul travelled $300 \ km$ by train and $200 \ km$ by taxi taking $5$ hours $30$ minutes. But, if he travels $260 \ km$ by train and $240 \ km$ by taxi, he takes $6$ minutes longer. Find the speed of the train and that of the taxi.
→Find the sum of all odd numbers between:
100 and 200.
100 and 200.
From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. $[ \text {Take} $ $\pi=\frac{22}{7}]$
→Marbles of diameter 1.4cm are dropped into a cylindrical beaker of diameter 7cm containing some water. Find the number of marbles that should be dropped into the beaker so that the water level rises by 5.6cm.