
Find the equation of the tangent to the parabola with gradient unity. Also find the area enclosed by the curve, the tangent line and
(i) the axis.
Answer
534.9k+ views
Hint: Try to imagine what the graph of would look like. This is also a quadratic in . So, it would most probably look like a bowl touching the axis at the origin, since the value of increases faster than the value of . Next, write the general equation of the tangent to this curve which is , and find the point where it touches the parabola. You will then get the limits of the integral you have to integrate.
Complete Step-by-Step Solution:
For a parabola of the general formula , the tangent to any point lying on the parabola can be written as : , where m is the slope of the tangent, and is thus, variable.
Comparing the parabola given to us in the question, to the general formula, we get :
Thus, the equation of a tangent to this parabola is : , where is the slope and is variable.
It is given to us in the question that the gradient of this tangent should be unity, and hence, the slope of the tangent should be equal to 1, or . Substituting for in the equation of the general tangent, we get :
Let’s call this equation as equation (1).
Now, let’s analyse the parabola given to us. It looks like a quadratic, where the coefficient of is positive. Hence, we know that the graph of this parabola will be concave upward. Since we can’t see any constant term in this equation, we can safely conclude that the graph passes through the origin definitely.
Satisfying the equation with the origin also confirms our assumption. However, since , we know that can be 0 at only one value, since both the roots of this equation will be 0 only.
Thus, we can now picture the graph better. It looks like a bowl, and touches the origin.
Now, let’s find the point where the tangent touches this parabola.
Substituting for , in terms of from (1), in the equation of the parabola, we get :
Therefore, the point where the tangent touches the parabola is (2,1).
Now let’s imagine the area being asked for. We need to find the area enclosed between the tangent, the parabola, and the axis. Keep in mind that the parabola touches the axis at the origin already.
Therefore, we can interpret the area being asked for in the following manner : it is the difference of the area enclosed by the parabola from to and the area enclosed by the line in the same interval.
In general, if is a curve lying above on the graph, then the area enclosed by the axis, and in the interval (a,b) =
Here, imagining will make us realise that the parabola lies above the tangent in the graph, and hence =parabola and =tangent with gradient unity, and the interval = (0,2).
Here’s a diagram to help you imagine the figure.
Hence, the required area =
Therefore, we have our required area, .
Then the required area = .
Note: Hence, from the diagram, you can clearly see that the tangent touches the parabola at (2,1), and the area asked for is indeed the difference of the area enclosed by the parabola and the tangent, in the interval (0,2).
Complete Step-by-Step Solution:
For a parabola of the general formula
Comparing the parabola given to us in the question, to the general formula, we get :
Thus, the equation of a tangent to this parabola is :
It is given to us in the question that the gradient of this tangent should be unity, and hence, the slope of the tangent should be equal to 1, or
Let’s call this equation as equation (1).
Now, let’s analyse the parabola given to us. It looks like a quadratic, where the coefficient of
Satisfying the equation with the origin also confirms our assumption. However, since
Thus, we can now picture the graph better. It looks like a bowl, and touches the origin.
Now, let’s find the point where the tangent touches this parabola.
Substituting for
Therefore, the point where the tangent touches the parabola is (2,1).
Now let’s imagine the area being asked for. We need to find the area enclosed between the tangent, the parabola, and the
Therefore, we can interpret the area being asked for in the following manner : it is the difference of the area enclosed by the parabola from
In general, if
Here, imagining will make us realise that the parabola lies above the tangent in the graph, and hence
Here’s a diagram to help you imagine the figure.

Hence, the required area =
Therefore, we have our required area,
Then the required area =
Note: Hence, from the diagram, you can clearly see that the tangent touches the parabola at (2,1), and the area asked for is indeed the difference of the area enclosed by the parabola and the tangent, in the interval (0,2).
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