What is the minimum area of the triangle formed by any tangent to the ellipse with the coordinate axis.
Answer
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Hint: The standard form equation of the ellipse is given by .The coordinates of any point lying on the ellipse is represented by . The equation of the tangent at this point on the ellipse is given by .
To minimise the area of the triangle, we have to find the generalised form of the area formed using the tangent and ellipse and determine the condition to minimize the area.
Complete step-by-step solution:
The given ellipse has the equation .
Choose any point R on the ellipse, so the coordinates of the point R are represented as .
Let us draw the tangent at this point R on the ellipse, such that the tangent meets the y-axis at point Q and meets x-axis at point P.
The given figure represents the above ellipse and the mentioned tangent:
Now, the parametric form of the equation of the tangent at the point is given by .
The above equation can be written as
So, the x and y coordinates of the above tangent equation are
Now, we get the three coordinates of the triangle formed by the tangent on the ellipse.
The triangle has the coordinates,
Find the area of the above triangle, using the formulae: , where base is and height is .
Area of
Since, and cannot be negative so modulus can be removed from them,
Area of
Use the identity:
Area of
To reduce the area of the triangle , we need to reduce the value of .
Since,
Taking the reciprocal,
Since, the minimum value is 1, hence the minimum value of the area of is .
The minimum area of the triangle formed by any tangent to the ellipse with the coordinate axis is .
Note: The minimum value of can be and the maximum value is . So, the value of always lies between and , and can be represented as, or .The equation of the form has intercept , and intercept is .
To minimise the area of the triangle, we have to find the generalised form of the area formed using the tangent and ellipse and determine the condition to minimize the area.
Complete step-by-step solution:
The given ellipse has the equation
Choose any point R on the ellipse, so the coordinates of the point R are represented as
Let us draw the tangent at this point R on the ellipse, such that the tangent meets the y-axis at point Q and meets x-axis at point P.
The given figure represents the above ellipse and the mentioned tangent:
Now, the parametric form of the equation of the tangent at the point
The above equation can be written as
So, the x and y coordinates of the above tangent equation are
Now, we get the three coordinates of the triangle formed by the tangent on the ellipse.
The triangle
Find the area of the above triangle, using the formulae:
Area of
Since,
Area of
Use the identity:
Area of
To reduce the area of the triangle
Since,
Taking the reciprocal,
Since, the minimum value
The minimum area of the triangle formed by any tangent to the ellipse
Note: The minimum value of
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