Animated Solution for Mathematics - Conic Sections: The equation of a tangent to the parabola $y^2=8x$ is $y=x+2$. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

Welcome, fellow explorer of mathematics. Today, we are not just solving a problem; we are uncovering a hidden symmetry in the world of parabolas.

Imagine you are standing on a coordinate plane, looking at the curve $y^2=8x$. It is a classic, right-opening parabola, elegant and simple.

Now, imagine a line, $y=x+2$, cutting across this space, grazing the parabola at a single point. This is our tangent. The problem asks us to find a point $P$ on this line such that if we were to draw a second tangent from $P$, it would be perfectly perpendicular to our first one.

The Hidden Truth

Many students immediately jump to the slope form of a tangent, $y=mx+\frac{a}{m}$, and start calculating. While that works, it is like walking a long, winding path when there is a shortcut right in front of you.

The shortcut is the Director Circle property. In the study of conics, the director circle is the locus of the intersection of perpendicular tangents.

For a parabola, this 'circle' has an infinite radius, which means it degenerates into a straight line: the directrix! This is a profound realization. It means that any two perpendicular tangents to a parabola must intersect on its directrix.

Our mystery point $P$ is not just any point; it is a point on the directrix.

Bringing It Home

Now that we have the key, the lock opens easily. First, we identify the parameter $a$.

Comparing $y^2=8x$ with the standard form $y^2=4ax$, we see that $4a=8$, which gives us:

The directrix of a right-opening parabola is the vertical line $x=-a$. Substituting our value, we get the equation of the directrix:

This is our second constraint. We know $P$ lies on the line $y=x+2$ and on the line $x=-2$.

To find the intersection, we simply substitute $x=-2$ into the tangent equation:

Thus, the coordinates of our point $P$ are $(-2,0)$.

Look at how beautiful that is. By understanding the geometric soul of the parabola, we bypassed the need for complex algebra or calculus.

We found that the point $(-2,0)$ is the unique location on the tangent where the perpendicularity condition is satisfied. Keep this property in your toolkit—it is a powerful weapon for the JEE Advanced.

Geometry is not just about shapes; it is about the relationships between them. Stay curious, keep visualizing, and the math will always reveal its secrets to you.