91Ó°ÊÓ

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 \(\Delta = 0\): \(\Delta = (2cm)^2 - 4 \cdot a^2 = 0\)

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 \(y=-\frac{a}{c}x\).

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 \((a, -\frac{a^2}{c})\).

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.