Question

\(f(x)\) is a continuous function on \(\mathbb{R}\) and \(y = f(x)\) is a curve. If \((\alpha, \beta)\) is a point such that \(\beta = f(\alpha)\) and \(p\alpha + m\beta + n = 0\ (p \ne 0, m \ne 0)\), then which one of the following is True?

• A. When \(p + mf'(\alpha) = 0\), the line \(px + my + n = 0\) intersects the curve \(y = f(x)\)
• B. \(px + my + n = 0\) is always a tangent to the curve \(y = f(x)\)
• C. When \(p + mf'(\alpha) \ne 0\), the line \(px + my + n = 0\) intersects the curve \(y = f(x)\)
• D. The line \(px + my + n = 0\) is never a tangent to the curve \(y = f(x)\)

Hint:

To check for tangency between a curve and a line at a point, ensure both the point lies on the curve and the slopes match. If the point lies but the slopes differ, it is an intersection, not a tangent.

Answer 1

The Correct Option is C

Given: \(f(x)\) is continuous and \((\alpha, \beta)\) lies on the curve \(y = f(x)\), with \(\beta = f(\alpha)\) and \(p\alpha + mf(\alpha) + n = 0\) This means the point \((\alpha, f(\alpha))\) lies on the line \(px + my + n = 0\). Now, for this line to be **tangent** to the curve at that point, it must also have the same slope as the curve at that point. Slope of the line: rearranged as \(y = -\frac{p}{m}x - \frac{n}{m}\), so slope is \(-\frac{p}{m}\) Slope of the curve at \((\alpha, f(\alpha))\): \(f'(\alpha)\) So, for tangency: \[ f'(\alpha) = -\frac{p}{m} \Rightarrow p + mf'(\alpha) = 0 \] Therefore, when \(p + mf'(\alpha) \ne 0\), the line is not tangent at that point. However, since the point \((\alpha, f(\alpha))\) lies on both the curve and the line, the line must **intersect** the curve at that point (though not tangentially).