Question

If two tangents drawn from a point P to the parabola \(y^2 = 16(x-3)\) are at right angles, then the locus of point P is :

• A. x + 1 = 0
• B. x + 2 = 0
• C. x + 3 = 0
• D. x + 4 = 0

Hint:

Remember this important property: The locus of the point of intersection of perpendicular tangents to a parabola is its directrix. This provides a direct and quick way to solve such problems without needing to use the general equation of tangents.

Answer 1

The Correct Option is A

Step 1: Identify the key property.
The locus of points from which two perpendicular tangents can be drawn to a parabola is its director circle. For a parabola, the director circle is a straight line, which is the directrix of the parabola.
Step 2: Find the equation of the directrix.
The given parabola is \(y^2 = 16(x-3)\). This is of the form \(Y^2 = 4aX\), where the vertex is shifted. - Let \(Y = y\) and \(X = x-3\). The vertex is at \(X=0, Y=0\), which means \(x=3, y=0\). - Comparing with the standard form, \(4a = 16\), so \(a = 4\). The equation of the directrix for the standard parabola \(Y^2 = 4aX\) is \(X = -a\). Now, we substitute back the original variables. \[ x - 3 = -4 \] \[ x = -1 \] This can be written as \(x + 1 = 0\). Step 3: Conclusion.
The locus of point P is the directrix of the parabola, which is the line \(x+1=0\).