Question

Find the number of common tangents to the curve $xy = {c^2}$ and ${y^2} = 4ax$.

Answer 1

In this question, use the equation of tangent with slope to obtain the slope of straight lines as the given curves of parabola and hyperbola. Find the point of contact and take discriminant value as zero to solve it.

Complete step-by-step solution:
We know that tangent on a point is a line or plane that touches the curved surface at a point, but if enlarge does not cross it at that point. The tangent gives the slope of a straight line.
The equation of tangent to the hyperbola is written as,
$xy = {c^2}......\left( 1 \right)$
Hyperbola is the open curve by cone which is in circular shape with a plane.
And,
Parabola is a curve of quadratic equations. It is a curve formed by a cone which is in circular shape with a parallel plane.
We know that the equation of tangent with slope $m$to the parabola ${y^2} = 4ax$ is written as,
$y = mx + \dfrac{a}{m}......\left( 2 \right)$
Now we substitute $mx + \dfrac{a}{m}$ for $y$ in equation (1) to find the point of contact on the hyperbola.
$x\left( {mx + \dfrac{a}{m}} \right) = {c^2}$
On simplification we get,
$\Rightarrow {m^2}{x^2} + ax - m{c^2} = 0$
Now by finding the point of contact and the roots of the equation, it can be observed that the number of points of contact is one, so in the equation there will be two equal roots.
The discriminant is equal to zero $\left( {D = 0} \right)$ and it is simplified as,
${a^2} + 4{m^3}{c^2} = 0$
On simplification we get,
$\therefore m = - {\left( {\dfrac{a}{{2c}}} \right)^{\dfrac{2}{3}}}$
Hence, it is observed that the slope $m$ has only one value, that is negative value.

Therefore, the number of common tangents will be one.

Note: As we know that a slope of line is a number that describes the direction and the slope of the line. Slope is the ratio of vertical change to the horizontal change between any two distinct points in line. It is also called a gradient of straight line.