The tangent and the normal drawn to the curve at cut the – axis at and respectively. If the length of the subtangent drawn to the curve at is equal to the length of the subnormal then the area of the triangle in sq. units is
Explanation for the correct option:
Step 1: Calculate the slope tangent
Given curve is,
The tangent to a curve can be found by calculating the derivative of the curve. Thus, the tangent of the given curve is,
The slope of the tangent at a given point can be calculated by substituting the value of -coordinate of the point into the equation of the tangent. So, the slope of the tangent at is,
Step 2: Determine the equation of the tangent
The equation of the tangent to the line can be determined using the two-point form of a line which is given by,
Here, and the slope
Thus, the equation of the tangent line is,
Step 3: Calculate the slope of the normal
The normal to a curve at a point is the line that is perpendicular to the tangent to the curve at that point.
We know that two lines are perpendicular if the sum of their product is .
Thus, the normal to a point of a curve is the negative reciprocal of the tangent, i.e.,
The slope of the normal is calculated the same way as that of a tangent.
Therefore, the slope of the normal to the curve at is,
Step 4: Determine the equation of the normal to the curve
The equation of the normal line can be determined using the two-point form of a line which is given by,
Here, and the slope
Thus, the equation of the tangent line is,
Step 5: Determine the coordinates of point
The equation of the tangent to the curve is (as derived earlier)
Given, point is the -intercept of the tangent.
This implies that the -coordinate of the point is .
Substituting this in the equation of the tangent, we get
Therefore, the coordinates of the point is .
Step 6: Determine the coordinates of point
The equation of the tangent to the curve is (as derived earlier)
Given, point is the -intercept of the tangent.
This implies that the -coordinate of the point is .
Substituting this in the equation of the tangent, we get
Therefore, the coordinates of the point is .
Step 7: Use the determinant method to find the area of the triangle
The coordinates of are , and of points , and respectively.
The area of a triangle when the coordinates of its three vertices are given is given by the formula,
where the first column is the coordinates of the vertices and the second column is their respective coordinates.
Thus, the area of is,
Therefore, the area of the is sq.units.
Hence, option D is the correct option.