Question

Plz Answer with explanation ​

Answer 1

Answér :

(ii) y = x² + 1

Note :

• If the equation of any curve remains unchanged when y is replaced by (-y) , then the curve is symmetrical about the x-axis .

Eg : y² = 4ax

• If the equation of any curve remains unchanged when x is replaced by (-x) , then the curve is symmetrical about the y-axis .

Eg : x² = 4ay

• If the equation of any curve remains unchanged when both x and y are replaced by (-x) and (-y) respectively then the curve is symmetrical in opposite coordinates ( ie. the curve is symmetrical about both the axes ) .

Eg : x² + y² = a²

• If the equation of any curve remains unchanged when x and y are interchanged , then the curve is symmetrical about the line x = y .

Eg : x³ + y³ = 3axy

Solution :

For checking the symmetricity of the curves about y-axis , replace x by (-x) . If the equation of the curve remains same , then the curve will be symmetrical about y-axis otherwise not symmetrical about y-axis .

• (i) y = (x + 1)² = x² + 2x + 1

Now ,

Replacing x by (-x) , we have ;

→ y = (-x)² + 2(-x) + 1

→ y = x² - 2x + 1

Clearly ,

The equation is changed , hence this curve is not symmetrical about y-axis .

• (ii) y = x² + 1

Now ,

Replacing x by (-x) , we have ;

→ y = (-x)² + 1

→ y = x² + 1

Clearly ,

The equation is unchanged , hence this curve is symmetrical about y-axis .

• (iii) y = (x - 1)² = x² - 2x + 1

Now ,

Replacing x by (-x) , we have ;

→ y = (-x)² - 2(-x) + 1

→ y = x² + 2x + 1

Clearly ,

The equation is changed , hence this curve is not symmetrical about y-axis .

• (iv) y² = x + 1

Now ,

Replacing x by (-x) , we have ;

→ y = x + 1

→ y = -x + 1

Clearly ,

The equation is changed , hence this curve is not symmetrical about y-axis .

Hence ,

The equation (ii) y = x² + 1 is symmetrical about y-axis .