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xx1 + yy1 = a2 is the equation of the
- a)Tangent to the circle x2+ y2=a2 at point (x1,y1)
- b)chord of contact of tangents drawn from the external point (x1, y1) to the circle x2+ y2 = a2
- c)polar of the point (x1, y1) w.r.t the circle x2 + y2 = a2
- d)all of the above
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
xx1 + yy1 = a2 is the equation of thea)Tangent to the circle x2+ y2=a2...
Remark : 1. The following three equations coincide:
(i) Equation of the tangent to a circle at a point P(x1, y1)
(ii) Equation of the chord of contact of the tangents to the circle from an outside point (x1, y1)
(iii) Equation of the polar of a point (x, y) inside or outside the circle (but not on the circle), with repsect to the circle.
Remark: 2. If the point P(x1, y1) is on the circle, then its polar coincides with the tangent at that point.
Remark: 3. If the polar of a point P passes through a point Q, then the polar of Q passes through P.
(i) Equation of the tangent to a circle at a point P(x1, y1)
(ii) Equation of the chord of contact of the tangents to the circle from an outside point (x1, y1)
(iii) Equation of the polar of a point (x, y) inside or outside the circle (but not on the circle), with repsect to the circle.
Remark: 2. If the point P(x1, y1) is on the circle, then its polar coincides with the tangent at that point.
Remark: 3. If the polar of a point P passes through a point Q, then the polar of Q passes through P.
Most Upvoted Answer
xx1 + yy1 = a2 is the equation of thea)Tangent to the circle x2+ y2=a2...
Remark : 1. The following three equations coincide:
(i) Equation of the tangent to a circle at a point P(x1, y1)
(ii) Equation of the chord of contact of the tangents to the circle from an outside point (x1, y1)
(iii) Equation of the polar of a point (x, y) inside or outside the circle (but not on the circle), with repsect to the circle.
Remark: 2. If the point P(x1, y1) is on the circle, then its polar coincides with the tangent at that point.
Remark: 3. If the polar of a point P passes through a point Q, then the polar of Q passes through P.
(i) Equation of the tangent to a circle at a point P(x1, y1)
(ii) Equation of the chord of contact of the tangents to the circle from an outside point (x1, y1)
(iii) Equation of the polar of a point (x, y) inside or outside the circle (but not on the circle), with repsect to the circle.
Remark: 2. If the point P(x1, y1) is on the circle, then its polar coincides with the tangent at that point.
Remark: 3. If the polar of a point P passes through a point Q, then the polar of Q passes through P.
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Community Answer
xx1 + yy1 = a2 is the equation of thea)Tangent to the circle x2+ y2=a2...
Explanation:
To understand why the correct answer is option 'D', let's break down each option and see why it is true.
a) Tangent to the circle x^2 + y^2 = a^2 at point (x1, y1):
The equation xx1 + yy1 = a^2 represents a line passing through the point (x1, y1). If this line is a tangent to the circle x^2 + y^2 = a^2, it means that the line intersects the circle at exactly one point. Therefore, option 'a' is valid.
b) Chord of contact of tangents drawn from the external point (x1, y1) to the circle x^2 + y^2 = a^2:
If we draw tangents from an external point (x1, y1) to the circle x^2 + y^2 = a^2, the points of contact will lie on a line called the chord of contact. The equation xx1 + yy1 = a^2 represents this line passing through the point (x1, y1). Hence, option 'b' is also valid.
c) Polar of the point (x1, y1) with respect to the circle x^2 + y^2 = a^2:
The polar of a point with respect to a circle is the line passing through the point and perpendicular to the line connecting the center of the circle and the point. In this case, the equation xx1 + yy1 = a^2 represents the polar of the point (x1, y1) with respect to the circle x^2 + y^2 = a^2. Thus, option 'c' is true.
d) All of the above:
We have already established that options 'a', 'b', and 'c' are valid. Therefore, the correct answer is option 'd' - all of the above.
In summary, the equation xx1 + yy1 = a^2 represents a line that can be interpreted as a tangent to the circle, the chord of contact of tangents from an external point, and the polar of a point with respect to the circle. Hence, all the options are correct.
To understand why the correct answer is option 'D', let's break down each option and see why it is true.
a) Tangent to the circle x^2 + y^2 = a^2 at point (x1, y1):
The equation xx1 + yy1 = a^2 represents a line passing through the point (x1, y1). If this line is a tangent to the circle x^2 + y^2 = a^2, it means that the line intersects the circle at exactly one point. Therefore, option 'a' is valid.
b) Chord of contact of tangents drawn from the external point (x1, y1) to the circle x^2 + y^2 = a^2:
If we draw tangents from an external point (x1, y1) to the circle x^2 + y^2 = a^2, the points of contact will lie on a line called the chord of contact. The equation xx1 + yy1 = a^2 represents this line passing through the point (x1, y1). Hence, option 'b' is also valid.
c) Polar of the point (x1, y1) with respect to the circle x^2 + y^2 = a^2:
The polar of a point with respect to a circle is the line passing through the point and perpendicular to the line connecting the center of the circle and the point. In this case, the equation xx1 + yy1 = a^2 represents the polar of the point (x1, y1) with respect to the circle x^2 + y^2 = a^2. Thus, option 'c' is true.
d) All of the above:
We have already established that options 'a', 'b', and 'c' are valid. Therefore, the correct answer is option 'd' - all of the above.
In summary, the equation xx1 + yy1 = a^2 represents a line that can be interpreted as a tangent to the circle, the chord of contact of tangents from an external point, and the polar of a point with respect to the circle. Hence, all the options are correct.
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xx1 + yy1 = a2 is the equation of thea)Tangent to the circle x2+ y2=a2 at point (x1,y1)b)chord of contact of tangents drawn from the external point (x1, y1) to the circle x2+ y2 = a2c)polar of the point (x1,y1) w.r.t the circle x2 + y2 = a2d)all of the aboveCorrect answer is option 'D'. Can you explain this answer?
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