CBSE Class 12-science Answered

Find the points on the curve 9y² = x³ where normal to the curve makes equal intercepts with the axes.

Asked by Topperlearning User | 07 Aug, 2014, 12:43: PM

Expert Answer

Let the required point be (x₁, y₁)

The equation of the curve is 9y² = x²

Since (x₁, y₁) lies on the curve. Therefore,

9 straight y subscript 1 squared equals straight x subscript 1 cubed...... left parenthesis straight i right parenthesis

Now, 9y² = x³

$rightwards double arrow dy over dx equals fraction numerator straight x squared over denominator 6 straight y end fraction rightwards double arrow open parentheses dy over dx close parentheses subscript left parenthesis straight x subscript 1 comma straight y subscript 1 right parenthesis end subscript equals fraction numerator straight x subscript 1 squared over denominator 6 straight y subscript 1 end fraction$

Since the normal to the curve at (x₁, y₁) make equal intercepts with the coordinate axes. Therefore,

Slope of the normal = ± 1

$rightwards double arrow minus 1 over open parentheses begin display style dy over dx end style close parentheses subscript left parenthesis straight x subscript 1 comma straight y subscript 1 right parenthesis end subscript equals plus-or-minus 1 rightwards double arrow open parentheses dy over dx close parentheses subscript left parenthesis straight x subscript 1 comma straight y subscript 1 right parenthesis end subscript equals plus-or-minus 1 rightwards double arrow fraction numerator straight x subscript 1 squared over denominator 6 straight y subscript 1 end fraction equals plus-or-minus 1 rightwards double arrow straight x subscript 1 to the power of 4 equals 36 straight y subscript 1 squared rightwards double arrow straight x subscript 1 to the power of 4 equals 36 open parentheses straight x subscript 1 cubed over 9 close parentheses left square bracket using space left parenthesis straight i right parenthesis right square bracket rightwards double arrow straight x subscript 1 to the power of 4 equals 4 straight x subscript 1 cubed rightwards double arrow straight x subscript 1 cubed left parenthesis straight x subscript 1 minus 4 right parenthesis equals 0 rightwards double arrow straight x subscript 1 equals 0 comma space 4$

Putting x₁ = 0 in (i), we get

9 straight y subscript 1 squared equals 0 rightwards double arrow straight y subscript 1 equals 0

Putting x₁ = 4 in (i), we get

9 straight y subscript 1 squared equals 4 cubed rightwards double arrow straight y subscript 1 equals plus-or-minus 8 over 3

But, the line making equal intercepts with the coordinate axes cannot pass through the origin.

Hence, the required points are _{$open parentheses 4 comma 8 over 3 close parentheses space & space open parentheses 4 comma minus 8 over 3 close parentheses$}