Solution:

Given equation is t^2 p^2=2(p 1)t.

Let's say that the roots of the equation are t1 and t2.

Coincident Roots

When the roots of the quadratic equation are coincident, it means that the equation has only one root. In other words, the discriminant of the quadratic equation is zero.

The discriminant of a quadratic equation ax^2 + bx + c = 0 is given by b^2 - 4ac.

If the discriminant is zero, the roots are coincident. If the discriminant is positive, the roots are real and distinct. If the discriminant is negative, the roots are complex.

Determining p

Let's substitute t1 and t2 with t in the given equation as both are coincident.

t^2 p^2=2(p 1)t
t^2 p^2 - 2(p+1)t = 0

Now, we can see that this is a quadratic equation in t. For this equation to have coincident roots, the discriminant must be zero.

So,

b^2 - 4ac = 0
(-2(p+1))^2 - 4(p^2) = 0
4p^2 + 8p + 4 - 4p^2 = 0
8p + 4 = 0
p = -1/2

Therefore, p = -1/2 will result in the given equation having coincident roots.