Simplify -9x2-y2-3x2-10xy-3y2

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Rational Algebraic Expression

Simplify :9x2...

Question

Simplify : $\frac{9 x^{2} - y^{2}}{3 x^{2} - 10 x y + 3 y^{2}}$

\frac{x + 3 y}{x - 3 y}

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\frac{3 x + y}{3 x - y}

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\frac{3 x - y}{x + 3 y}

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\frac{3 x + y}{x - 3 y}

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Solution

The correct option is D $\frac{3 x + y}{x - 3 y}$
Given expression: $\frac{9 x^{2} - y^{2}}{3 x^{2} - 10 x y + 3 y^{2}}$

As $9 x^{2} - y^{2}$ and $3 x^{2} - 10 x y + 3 y^{2}$ are two algebraic expressions and are in the form of $\frac{A}{B}$ , $\frac{9 x^{2} - y^{2}}{3 x^{2} - 10 x y + 3 y^{2}}$ is a rational algebraic expression.

Consider the numerator i.e. $9 x^{2} - y^{2}$
It can be written as $(3 x)^{2} - (y)^{2}$
It is in the form of $a^{2} - b^{2}$
We know that $a^{2} - b^{2} = (a - b) (a + b)$
$\Rightarrow$ $(3 x)^{2} - (y)^{2} = (3 x - y) (3 x + y)$

Consider the denominator i.e. $3 x^{2} - 10 x y + 3 y^{2}$
$3 x^{2} - 10 x y + 3 y^{2}$ can be written as
$3 x^{2} - 9 x y - x y + 3 y^{2}$
$\Rightarrow$ $3 x (x - 3 y) - y (x - 3 y)$
$\Rightarrow$ $(3 x - y) (x - 3 y)$

Now, $\frac{9 x^{2} - y^{2}}{3 x^{2} - 10 x y + 3 y^{2}} = \frac{(3 x - y) (3 x + y)}{(3 x - y) (x - 3 y)}$

$= \frac{3 x + y}{x - 3 y}$
$∴ \frac{9 x^{2} - y^{2}}{3 x^{2} - 10 x y + 3 y^{2}} = \frac{3 x + y}{x - 3 y}$

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Similar questions

$Find the ecentrictity, coordinates of foci, equations of directrices and length of the latus-rectum of the hyperbola (i) 9 x^{2} - 16 y^{2} = 144 (i i) 16 x^{2} - 9 y^{2} = - 144 (i i i i) 4 x^{2} - 3 y^{2} = 36 (i v) 3 x^{2} - y^{2} = 4 (v) 2 x^{2} - 3 y^{2} = 5$

Q. Multiply:
(3x² + y²) by (2x² + 3y²)

Q. Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus-rectum of the hyperbola
(i) 9x² − 16y² = 144
(ii) 16x² − 9y² = −144
(iii) 4x² − 3y² = 36
(iv) 3x² − y² = 4
(v) 2x² − 3y² = 5.

Q. Assertion :Angle between the lines represented by

3 x^{2} + 10 x y + 3 y^{2} - 2 x + 2 y - 1 = 0

and

3 x^{2} + 10 x y + 3 y^{2} - 11 x + 7 y - 20 = 0

is same. Reason: Equations of the bisectors of the angles between the lines

3 x^{2} + 10 x y + 3 y^{2} = 0

x^{2} - y^{2} = 0

If the pairs of lines $x^{2} + 2 x y + a y^{2} = 0$ and $a x^{2} + 2 x y + y^{2} = 0$ have exactly one line in common then the joint equation of the other two lines is given by

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Simplify :9x2−y23x2−10xy+3y2

Simplify : $\frac{9 x^{2} - y^{2}}{3 x^{2} - 10 x y + 3 y^{2}}$