Since the x-intercept of the line is (10,0), we know that the equation of the line is of the form y=mx+n for some value of m and n. Since the y-intercept of the line is (0,−2), we have that −2=m⋅0+n. Hence, n=−2. Therefore, the equation of the line is y=mx−2.

We know that the point (a,a2) lies on the line, so a2=ma−2. We also know that the point (b,b2) lies on the line, so b2=mb−2. Subtracting these two equations, we get b2−a2=m(b−a). Since a 0, so m=b−ab2−a2.

Since the line passes through the point (10,0), we have that 0=m(10)−2. Substituting the expression for m into this equation, we get 0=b−ab2−a2(10)−2. This simplifies to b2−a2=5a−5b. We also have the equation a2=ma−2. Substituting m=b−ab2−a2, we get a2=b−ab2−a2(a)−2. This simplifies to (b−a)a2−(b2−a2)a+2(b−a)=0. This factors as (a−2)(b−a)a2=0.

Since a

Substituting a=0 into the equation b2−a2=5a−5b, we get b2=5b. This factors as b2−5b=0, which factors as b(b−5)=0. Therefore, b=0 or b=5. Since a

Therefore, (a,b)=(0,5).