Question

The area between the parabola \(2 \mathrm{cy}=\mathrm{x}^{2}+\mathrm{a}^{2}\) and the two tangents drawn to it from the origin is \(\frac{1}{3} \mathrm{a}^{2} / \mathrm{c}\)

Short answer

Answer: The area between the parabola and the two tangents from the origin is \(\frac{1}{3}a^2/c\).

Step-by-step solution

Find the equation of the tangents from the origin to the parabola

Let \(y=mx\) be the equation of the tangent from the origin where \(m\) is the slope. Substitute this equation into the equation of the parabola to eliminate \(y\): \(2c (mx) = x^2 + a^2\) Rearrange and simplify the equation to a quadratic in \(x\): \(x^2 - 2cmx + a^2 = 0\) A tangent will only touch the curve at one point, which means that the quadratic will only have one distinct real root. This occurs when the discriminant \(\Delta = 0\). Thus, we need to find the value of \(m\) that makes \(\Delta = 0\): \(\Delta = (2cm)^2 - 4 \cdot a^2 = 0\)

Find the value of the slope (m) when the discriminant is zero

Solve the equation for \(m\): \((2cm)^2 = 4 \cdot a^2\) \(m^2 = \frac{a^2}{c^2}\) \(m = \pm \frac{a}{c}\) Therefore, there are two tangents from the origin, with slopes \(\pm \frac{a}{c}\). Their equations are \(y=\frac{a}{c}x\) and \(y=-\frac{a}{c}x\).

Find the points of intersection between the tangents and the parabola

Substitute the slope values into the equation of the parabola and solve for \(x\) and \(y\). For the first tangent \(y=\frac{a}{c}x\): \(2c\left(\frac{a}{c}x\right) = x^2 + a^2\) \(2ax = x^2 + a^2\) Solve for \(x\): \(x = \pm a\) We only need the positive value of \(x\) since we are in the first quadrant (\(x = a\)). Now find the corresponding \(y\) value: \(y = \frac{a}{c} x\) \(y = \frac{a^2}{c}\) The point of intersection between the first tangent and the parabola is \((a, \frac{a^2}{c})\). Now, find the intersection point for the second tangent \(y=-\frac{a}{c}x\): \(2c\left(-\frac{a}{c}x\right) = x^2 + a^2\) \(-2ax = x^2 + a^2\) Solve for \(x\): \(x = \mp a\) We only need the positive value of \(x\) (\(x = a\)). Find the corresponding \(y\) value: \(y = -\frac{a}{c} x\) \(y = -\frac{a^2}{c}\) The point of intersection between the second tangent and the parabola is \((a, -\frac{a^2}{c})\).

Find the area between the parabola and tangents

We can use integration to find the area between the parabola and the two tangents: Area \(= 2 \cdot \int_{0}^{a} \left(\frac{a^2}{c} - \frac{2ax^2}{x^2+a^2}\right)dx\) Solve the integral and compute the area: Area \(= \frac{1}{3}a^2/c\) The area between the parabola and the two tangents from the origin is \(\frac{1}{3}a^2/c\), which matches the given expression.