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The gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2, is
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The gradient of the tangent line at the point (acosa,bcosa) to the cir...
Explanation of the Gradient of the Tangent Line to the Circle x^2 + y^2 = a^2
Definition of the Circle and the Tangent Line
The given circle equation is x^2 + y^2 = a^2, where a is the radius of the circle. Let P be a point on the circle with coordinates (acosa, bcosa), where b is the y-coordinate of the point. The tangent line at P is the line that touches the circle at P and has the same slope as the circle at that point.
Derivation of the Gradient of the Tangent Line
To find the gradient of the tangent line at P, we need to find the derivative of the circle equation with respect to x, evaluated at x = acosa. This is because the slope of the tangent line is equal to the derivative of the equation of the circle at that point.
We start by differentiating both sides of the circle equation with respect to x:
2x + 2y(dy/dx) = 0
dy/dx = -x/y
Next, we substitute x = acosa and y = bcosa into the derivative:
dy/dx = -acosa/bcosa
Therefore, the gradient of the tangent line at P is:
m = -acosa/bcosa
Interpretation of the Gradient of the Tangent Line
The negative sign indicates that the tangent line is sloping downwards to the right of the point P. The magnitude of the gradient is the tangent of the angle between the tangent line and the horizontal axis, which is equal to the inverse of the slope of the radius passing through P. This means that the gradient of the tangent line is steeper when the radius passing through P is flatter, and vice versa.
Conclusion
In summary, the gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2 is -acosa/bcosa. This represents the slope of the tangent line at that point and can be interpreted as the rate of change of y with respect to x at that point.
Community Answer
The gradient of the tangent line at the point (acosa,bcosa) to the cir...
-(a/b) bcoz eqn of tangent to a circle is xx1+yy1=a².
here x1 is equal to a cos a and y1 is equal to b cos a. Substituting these in the eqn, we get the slope to be -(a/b).
here x1 is equal to a cos a and y1 is equal to b cos a. Substituting these in the eqn, we get the slope to be -(a/b).
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The gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2, is
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The gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2, is for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about The gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2, is covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2, is.
The gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2, is for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about The gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2, is covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The gradient of the tangent line at the point (acosa,bcosa) to the circle x^2 + y^2 = a^2, is.
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