Question #209e6

1 Answer
Sep 13, 2016

Answer:

See the Explanation.

Explanation:

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.!