In this question we need to find the locus of intersection of two tangents. I am not getting it

To find the locus of intersection of two tangents to a curve, we first need to understand what a tangent is and how it relates to the curve. Let's break this down step by step.

Understanding Tangents

A tangent to a curve at a given point is a straight line that just touches the curve at that point. It has the same slope as the curve at that point. When we talk about two tangents, we usually refer to tangents drawn from an external point to the curve.

Setting Up the Problem

Consider a standard curve, such as a circle or a parabola. For simplicity, let’s take a circle with the equation:

x² + y² = r²

Now, if we have a point outside this circle, say (h, k), we can draw two tangents from this point to the circle. The goal is to find the locus of the intersection points of these tangents as the point (h, k) moves around.

Finding the Tangent Equations

The equation of the tangents from the point (h, k) to the circle can be derived using the formula:

y - k = m(x - h)

where m is the slope of the tangent. The tangents will intersect the circle at exactly one point, which means we can substitute this line equation into the circle's equation and solve for m. This will give us the slopes of the tangents.

Intersection of the Tangents

Once we have the equations of the tangents, we can find their intersection point. The intersection of two lines can be found by solving the two linear equations simultaneously. The coordinates of the intersection point will depend on the slopes and the point (h, k).

Finding the Locus

As the point (h, k) moves, the intersection point of the tangents will trace out a path. To find this path, we eliminate the parameters (h and k) from the equations of the tangents. This process typically involves substituting one equation into another or using algebraic manipulation to express the relationship between x and y directly.

Example with a Circle

For the circle x² + y² = r², if we derive the equations of the tangents and find their intersection, we can show that as (h, k) varies, the locus of the intersection points will be another geometric figure. In this case, it turns out to be another circle or a conic section, depending on the specific conditions.

Final Thoughts

In summary, to find the locus of intersection of two tangents, you need to:

• Identify the curve and the external point.
• Derive the equations of the tangents from that point.
• Find the intersection of those tangents.
• Eliminate parameters to find the locus as a function of x and y.

This process can be applied to various curves, and the resulting locus will depend on the nature of the original curve and the position of the external point. If you have a specific curve in mind, we can work through that example together!