Problem
Let P be a point on the parabola y² = 4ax, where a > 0. The normal to the parabola at P meets the x-axis at a point Q. The area of the triangle PFQ, where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then find the pair (a, m).
Solution
Let’s start by drawing things out.
We have two variables here, a and m. And two conditions to find their values.
One, the area of triangle PFQ = 120. And the second, both a and m are integers.
Let’s start by finding the area of PFQ in terms of a and m, and equating it to 120.
Since we’re dealing with the slope of the normal, we’ll use the slope form of its equation.
That is, the equation of the normal at P(am², -2am) is given by y = mx – 2am – am³.
This intersects the x-axis at (2a + am², 0). And, the point F has the coordinates (a, 0).
Now, the area of the triangle PFQ equals 1/2 x FQ x h, where h is its height or the distance of P from the x-axis.
Substituting the values, we’ll get the area as 1/2 x |(a + am²) x (-2am)|, or a²m(1 + m²).
Given that this equals 120, we have a²m(1 + m²) = 120.
Now, this equation has infinitely many solutions. But we need the solutions that are both integers.
Things get easier because of the a² term – there are limited ways in which 120 can be factorized where one of the factors is a perfect square.
Two ways, to be precise: 120 = 1² x 120 and 120 = 2² x 30
From the first one, we get a = 1, and m(1 + m²) = 120, which doesn’t lead to any integral solutions for m.
From the second one, we’ll get a = 2, and m(1 + m²) = 30, which gives m = 3.
So the required pair (a, m) is (2, 3).
Comments
The problem isn’t too difficult, if you’re familiar with the equation of a normal and the area of a triangle (which you have to be, given it’s the JEE Advanced!).
Secondly, once you have the final equation ready, you don’t really need to find out a and m as we did above. Instead, you could’ve easily substituted each answer option to figure out the correct one. So, yeah, this kind of ruined all the fun.
This should ideally have been posed as a slightly different problem, asking for the number of possible positive integral pairs (a, m).