Question

Which one of the following statements is incorrect?

• A. A point that separates the convex part of a continuous curve from the concave part is called a point of inflection.
• B. The equation of the asymptote to the curve \(y^2x^2 = x^2 - 1\) is \(y = \pm 1\).
• C. The curve \(x^3 - y^3 = 3xy\) is symmetrical about the line \(y = -x\).
• D. The equation of the tangent to the curve \(x^2y^2 = x^2 - 1\) at the point \((1, 0)\) is \(y = x + 1\).

Hint:

Check the tangent slope of \(x^2y^2=x^2-1\) at \((1,0)\) by implicit differentiation.

Answer 1

The Correct Option is D

Step 1: Option (A) states the standard definition of a point of inflection, a point where the curve changes from convex to concave or vice versa. This is a correct statement.

Step 2: For option (B), the curve is \(y^2x^2=x^2-1\), so \(y^2 = 1-\dfrac{1}{x^2}\), defined for \(|x|\ge 1\). As \(x\to\pm\infty\), \(y^2\to 1\), so \(y\to\pm1\). Hence \(y=1\) and \(y=-1\) are the horizontal asymptotes, matching \(y=\pm1\). This statement is correct.

Step 3: For option (C), the curve is \(x^3-y^3=3xy\). Reflecting a point \((x,y)\) in the line \(y=-x\) sends it to \((-y,-x)\). Substituting: \[ (-y)^3-(-x)^3 = 3(-y)(-x) \implies x^3-y^3 = 3xy \] which is exactly the original equation. So the curve maps to itself under this reflection, confirming symmetry about \(y=-x\). This statement is correct.

Step 4: For option (D), the curve is \(x^2y^2=x^2-1\). Differentiating implicitly: \[ 2xy^2 + 2x^2yy' = 2x \] At the point \((1,0)\), substitute \(x=1, y=0\): \[ 2(1)(0)+2(1)^2(0)y' = 2(1) \implies 0 = 2 \] This is a contradiction, which means \(dy/dx\) does not exist as a finite number at \((1,0)\); the tangent there is vertical, not the slanted line \(y=x+1\). So option (D) is false.

Step 5: Since (A), (B) and (C) are all true and (D) is false, the incorrect statement is (D). \[ \boxed{\text{Option (D) is incorrect}} \]