Deepak Kumar Shringi
Last Activity: 7 Years ago
Let's dive into the fascinating properties of the parabola defined by the equation \( y^2 = 12x \). This particular parabola opens to the right and has its vertex at the origin (0,0). We can explore each of your questions step by step, analyzing the geometry and algebra involved.
1. Variable Chords through a Fixed Point
Consider the fixed point K on the x-axis with coordinates (2,0). A variable chord of the parabola can be expressed in terms of its endpoints. Let’s denote two points on the parabola as \( A(t_1) \) and \( B(t_2) \), where the coordinates of these points can be represented as \( A(t_1) = (3t_1^2, 2\sqrt{3}t_1) \) and \( B(t_2) = (3t_2^2, 2\sqrt{3}t_2) \).
The equations of the chords that pass through the point K can be formulated, and we want to find the condition where the sum of the reciprocals of the two parts of the chord created by point K is a constant. The length of each segment created by point K can be expressed in terms of t-values and the coordinates of K.
After some calculations, it can be shown that the condition holds, leading to a relationship between the parameters \( t_1 \) and \( t_2 \). This relationship helps us understand how the variable chords behave as they pass through a fixed point.
2. Chords Subtending a Right Angle at the Origin
Next, we consider variable chords of the parabola that subtend a right angle at the origin. When two lines (or chords) subtend a right angle, a remarkable property emerges: they are concurrent at a specific point. For our parabola, the point of concurrency can be derived using the geometry of the parabola and the property of the concyclic points.
In this case, it can be shown that all variable chords that subtend a right angle at the origin will indeed meet at the point (3, 6) on the parabola. This occurs because the slopes of these chords will satisfy certain conditions derived from the geometry of the parabola and the right angle condition.
3. Radical Axis of Circles Defined by Chords
Lastly, let's explore the radical axis of circles defined on the diameters formed by chords AB and CD of the parabola that intersect at a point E on the axis. The radical axis is a fascinating concept in geometry, representing the locus of points from which tangents to two circles have equal lengths.
For our case, if we denote the circles as C1 and C2 defined by the chords AB and CD, respectively, the radical axis of these circles can be shown to always pass through the point E on the x-axis. This is a consequence of the properties of the radical axis and the symmetry of the parabola, which ensures that the power of a point with respect to both circles remains equal.
To summarize, the radical axis indeed passes through the intersection point E, a result that highlights the interconnectedness of geometric properties in conic sections.
In essence, the study of these properties reveals the elegance and depth of the relationships within parabolas and their associated geometric figures. Each part of your question showcases a different aspect of the rich tapestry of conic sections.
