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.!
