CBSE Class 12-science Answered

Find the point on the curve y = x³ – 11x + 5 at which the tangent has the equation y = x – 11.

Asked by Topperlearning User | 11 Aug, 2014, 10:06: AM

Expert Answer

Let the required point be P(x₁, y₁), since (x₁, y₁) lies on y = x³ – 11x + 5.

y₁ =

x subscript 1 cubed minus 11 x subscript 1 plus 5

…(i)

open parentheses fraction numerator d y over denominator d x end fraction close parentheses subscript left parenthesis x subscript 1 comma y subscript 1 right parenthesis end subscript equals

Now, y = x³ – 11x + 5

straight y equals straight x cubed minus 11 straight x plus 5 rightwards double arrow dy over dx equals 3 straight x squared minus 11 rightwards double arrow open parentheses dy over dx close parentheses subscript open parentheses straight x subscript 1 comma straight y subscript 1 close parentheses end subscript equals 3 straight x subscript 1 squared minus 11

Since the line y = x – 11 is tangent at the point (x₁, y₁). Therefore,

Slope of the tangent at (x₁, y₁) = (Slope of the line y = x – 11).

open parentheses dy over dx close parentheses subscript open parentheses straight x subscript 1 comma straight y subscript 1 close parentheses end subscript equals

(Slope of the line x – y – 11 = 0)

rightwards double arrow 3 straight x subscript 1 squared minus 11 equals fraction numerator minus 1 over denominator minus 1 end fraction open square brackets because Slope space equals minus fraction numerator coeff. space of space straight x over denominator coeff. space of space straight y end fraction close square brackets rightwards double arrow 3 straight x subscript 1 squared equals 12 rightwards double arrow straight x subscript 1 squared equals 4 rightwards double arrow straight x subscript 1 equals plus-or-minus 2

Now,

x subscript 1 equals 2 rightwards double arrow y equals 2 cubed minus 11 left parenthesis 2 right parenthesis plus 5 rightwards double arrow y equals minus 9

[Using (i)]

straight x subscript 1 equals minus 2 rightwards double arrow straight y equals left parenthesis minus 2 right parenthesis cubed minus 11 left parenthesis minus 2 right parenthesis plus 5 rightwards double arrow straight y equals 19

[Using (i)]

So, two points are (2, –9) and (–2, 19). Of these two points (–2, 19) does not lie on y = x – 11. Therefore, the required point is (2, – 9).

Answered by | 11 Aug, 2014, 12:06: PM

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