We use Reductio Ad Absurdum , or, the Method of Contradiction.
Suppose, to the contrary , that
EE" a line, say "l," with Y-intercept "10" and touching the Curve"
"(Parabola) C : "y=3x^2+7x-2.
Since, l touches C, l nnC must be a Singleton sub RR^2.
If the slope of l is m, then, the eqn. of l is y=mx+10.
[A Clarification : In case, m does not exist, then, l has to be
vertical, i.e., l || Y-Axis; so, l does not intersect Y-Axis, &, as such,
l" can not have "Y"-intercept"=10". Evidently, "m does exist. ]
To find l nnC, we solve their eqns.
y=mx+10, y=3x^2+7x-2 rArr mx+10=3x^2+7x-2.
:. 3x^2+(7-m)x-12=0..................(star)
In order that l nn C be Singleton, the qudr. eqn.(star) must
have two identical roots, for which Delta=0.
:. (7-m)^2-4(3)(-12)=0
(7-m)^2=-144, Impossible in RR.
This contradiction shows that our supposition is wrong.
Thus, no line having Y-intercept 10 can be tangential to
the given Parabola.
Enjoy Maths.!
