Question

Assertion :If lengths of sub tangent & subnormal at point (x, y) on y= f(x) are respectively 16 & 9 then $x=\pm 12$ Reason: Product of sub tangent & subnormal is square of the ordinates of the points.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct.

Answer 1

The correct option is D Assertion is incorrect but Reason is correct.

Let the tangent & normal at a point P(x,y) on the curve y=f(x) meet x-axis at T & N respectively. If G is the foot of the ordinate at P, then TG, GN, are called the Cartesian subtangent & subnormal, while the lengths PT & PN are called Lengths of tangent & normal respectively
If PT makes angle $\psi$ with x-axis then $\tan \psi =\frac{dy}{dx}$ and subtangent=TG$=y\cot \psi =\frac{y}{dy/dx}=\frac{y}{m}$ (i)
Also subnormal=GN$=y\tan \psi$
$=y\frac{dy}{dx}=ym$ (ii)
Now $\frac{y}{m}=16$ & $ym=9$ $\therefore$ $\frac{y}{m}.ym=16\times 9=144$
$y=\pm 12$
$\therefore$ Assertion is false
And product of subtangent & subnormal is $y^2$, so Reason (R) is true.