Distance Of A Point From A Line

We are given the straight line: and the equation of the circle: The chord of contact of tangents drawn from any point on the straight line to the circle is given by the equation: where is the equation of the tangent to the circle. The equation of the tangent to the circle at is: Now, we want the locus of the mid-point of the chord of contact, i.e., the mid-point of the tangents from points on the line . For the point lying on the line, we substitute and into the equation of the line: Now, using the fact that the mid-point of the chord of contact is given by the formula: After solving the resulting equations and simplifying, we find that the equation of the locus of the mid-point is: Thus, the correct answer is Option A.

The equation of the given parabola is: The slope of the tangent to the parabola at any point on the parabola is found by differentiating the equation implicitly: So, the slope of the tangent at is . The slope of the normal at this point is the negative reciprocal of the tangent slope: Now, we are given that the normal is inclined at with the horizontal line , which has a slope of 0. The angle between the normal and the horizontal line is given by: Thus, the slope of the normal can also be expressed as . Therefore: Solving for : Substitute this value of into the equation of the parabola: So, the point is . Now, to find the straight line that passes through this point, we substitute and into the equation of the straight line. The line that passes through this point is: Thus, the correct answer is Option C.

To determine the locus of point , we need to equate its distance from point to its distance from the line .

The distance formula from point to is given by:

The distance from point to the line is given by:

Given , we have:

Squaring both sides results in:

Expanding and simplifying gives:

Breaking down:

Rearranging terms:

Simplifying gives:

By completing the square and further simplification (not fully expanded here for brevity), we identify the equation as that of a parabola.

Thus, the locus of point is a parabola.