The value of $x^{2} - y^{2}$ , if $7 x + 8 y = 5$ and $8 x + 7 y = 10$ will be

Open in App

Solution

Given
$7 x + 8 y = 5$ $. . . . . (1)$
and $8 x + 7 y = 10$ $. . . . . . . . (2)$

Squaring equation $(1)$ and $(2)$ both side we get,
$(7 x + 8 y)^{2} = 5^{2}$

$\Rightarrow (7 x + 8 y)^{2} = 25$ $. . . . . . (3)$

Also,
$(8 x + 7 y)^{2} = 10^{2}$

$\Rightarrow (8 x + 7 y)^{2} = 100$ $. . . . . . (4)$

Substracting equation (3) from (4) we get,
$(8 x + 7 y)^{2} - (7 x + 8 y)^{2} = 100 - 25$

By using the formula $(a - b) (a + b) = (a^{2} - b^{2})$ we get,

$\Rightarrow [(8 x + 7 y) - (7 x + 8 y)] [(8 x + 7 y) + (7 x + 8 y)] = 75$

$\Rightarrow [8 x + 7 y - 7 x - 8 y] [8 x + 7 y + 7 x + 8 y] = 75$

$\Rightarrow (x - y) (15 x + 15 y) = 75$

$\Rightarrow 15 (x - y) (x + y) = 75$

$\Rightarrow 15 (x^{2} - y^{2}) = 75$

$\Rightarrow x^{2} - y^{2} = 5.$

Suggest Corrections

Similar questions

Find the equation of family of circles through the intersection of $x^{2} + y^{2} - 6 x + 2 y + 4 = 0$ and $x^{2} + y^{2} + 2 x - 4 y - 6 = 0$ whose center lies on $y = x .$

Q. The equation of the parabola with focus (0, 0) and directrix x + y = 4 is
(a) x² + y² − 2xy + 8x + 8y − 16 = 0
(b) x² + y² − 2xy + 8x + 8y = 0
(c) x² + y² + 8x + 8y − 16 = 0
(d) x² − y² + 8x + 8y − 16 = 0

Q. If

8 x + 8 y = 18

and

x^{2} - y^{2} = - \frac{3}{8}

, calculate the value of

2 x - 2 y

Q. The equation of the parabola whose vertex is

(- 3, 0)

and directrix is

x + 5 = 0,

Q. Find the values of

(x + y)

and

(x - y)

without actually solving for

x

and

y

8 x - 7 y = 11; 7 x - 8 y = 4

Join BYJU'S Learning Program

The value of x2−y2, if 7x+8y=5 and 8x+7y=10 will be

The value of $x^{2} - y^{2}$ , if $7 x + 8 y = 5$ and $8 x + 7 y = 10$ will be