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In an ellipse the locus of point of intersection of the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact, is
- a)corresponding directrix
- b)latus rectum
- c)director circle
- d)none of these
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
In an ellipse the locus of point of intersection of the perpendicular ...
Any point on the ellipse 
can be taken as P(a cosθ, b sinθ)
The equation of tangent at point P is
The equation of line perpendicular to tangent is
Since, the equation (i) passes through the focus (ae,0), then we get,
Thus, the equation (i) becomes
Now, the equation of line joining centre and point of contact (a cos θ, b sin θ) is
Solving the equation (ii) & (iii), we get the point of intersection Q whose abscissa is a/e.
Hence, Q lies on the corresponding directrix x = a/e.
can be taken as P(a cosθ, b sinθ)The equation of tangent at point P is
The equation of line perpendicular to tangent is
Since, the equation (i) passes through the focus (ae,0), then we get,
Thus, the equation (i) becomes
Now, the equation of line joining centre and point of contact (a cos θ, b sin θ) is
Solving the equation (ii) & (iii), we get the point of intersection Q whose abscissa is a/e.
Hence, Q lies on the corresponding directrix x = a/e.
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In an ellipse the locus of point of intersection of the perpendicular ...
Understanding the Locus in an Ellipse
In an ellipse, certain geometric properties lead to interesting loci of points related to its foci and tangents. Let's explore why the locus of the intersection of the perpendicular from a focus to any tangent and the line connecting the center to the point of contact corresponds to the directrix.
Key Concepts:
- Ellipse Definition: An ellipse is defined as the set of all points where the sum of the distances from two fixed points (foci) is constant.
- Focus and Directrix: Each focus has an associated directrix. For an ellipse, the directrix serves as a guiding line that helps define the shape and properties of the ellipse.
Geometric Construction:
- Tangent Line: At any point on the ellipse, you can draw a tangent line.
- Perpendicular from Focus: From a focus, you can drop a perpendicular line to the tangent.
- Connecting Line: The line that connects the center of the ellipse (the midpoint between the foci) to the point where the tangent touches the ellipse is also drawn.
Locus of Intersection:
- Intersection Point: The intersection of the perpendicular from the focus and the line from the center to the point of contact is of interest.
- Resulting Locus: As the tangent moves around the ellipse, the intersection point traces out a straight line, which is precisely the directrix associated with the corresponding focus.
Conclusion:
- The intersection point's trajectory aligns with the directrix, confirming that the correct answer to the question is indeed option 'A', the corresponding directrix. This relationship showcases the elegant symmetry and properties inherent in conic sections, particularly ellipses.
In an ellipse, certain geometric properties lead to interesting loci of points related to its foci and tangents. Let's explore why the locus of the intersection of the perpendicular from a focus to any tangent and the line connecting the center to the point of contact corresponds to the directrix.
Key Concepts:
- Ellipse Definition: An ellipse is defined as the set of all points where the sum of the distances from two fixed points (foci) is constant.
- Focus and Directrix: Each focus has an associated directrix. For an ellipse, the directrix serves as a guiding line that helps define the shape and properties of the ellipse.
Geometric Construction:
- Tangent Line: At any point on the ellipse, you can draw a tangent line.
- Perpendicular from Focus: From a focus, you can drop a perpendicular line to the tangent.
- Connecting Line: The line that connects the center of the ellipse (the midpoint between the foci) to the point where the tangent touches the ellipse is also drawn.
Locus of Intersection:
- Intersection Point: The intersection of the perpendicular from the focus and the line from the center to the point of contact is of interest.
- Resulting Locus: As the tangent moves around the ellipse, the intersection point traces out a straight line, which is precisely the directrix associated with the corresponding focus.
Conclusion:
- The intersection point's trajectory aligns with the directrix, confirming that the correct answer to the question is indeed option 'A', the corresponding directrix. This relationship showcases the elegant symmetry and properties inherent in conic sections, particularly ellipses.
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In an ellipse the locus of point of intersection of the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact, isa)corresponding directrixb)latus rectumc)director circled)none of theseCorrect answer is option 'A'. Can you explain this answer?
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