The length of sub-normal at any point P ( x , y ) on the curve, which is passing through M (0,1) is unity. The area bounded by the curves satisfying this condition is equal to

Question

The length of sub-normal at any point P ( x , y ) on the curve, which is passing through M (0,1) is unity. The area bounded by the curves satisfying this condition is equal to (A) (1/3) (B) (2/3) (C)  (4/3) (D) (8/3)

Tardigrade Admin · Accepted Answer

<img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/582d6512fbfee8cc3d75b4d9d144e8c2-.png" /> We have |( ydy / dx )|=1 ⇒ ∫ ydy =∫ ± dx ⇒ ( y 2/2)= ± x + C But M (0,1) satisfy it, so C =(1/2) ⇒ y2= ± 2 x +1 Let C1: y2=2(x+(1/2)) and C2: y2=-2(x-(1/2)) Clearly required area =4 ∫ limits(-1/2)0 √2 x +1 dx =(4/3) (square units)

Q. The length of sub-normal at any point $P (x, y)$ on the curve, which is passing through $M (0, 1)$ is unity. The area bounded by the curves satisfying this condition is equal to

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{4}{3}$

$\frac{8}{3}$

Q. The length of sub-normal at any point P(x,y) on the curve, which is passing through M(0,1) is unity. The area bounded by the curves satisfying this condition is equal to

31​

32​

34​

38​

We have ∣∣​dxydy​∣∣​=1⇒∫ydy=∫±dx⇒2y2​=±x+C But M(0,1) satisfy it, so C=21​ ⇒y2=±2x+1 Let C1​:y2=2(x+21​) and C2​:y2=−2(x−21​) Clearly required area =42−1​∫0​2x+1​dx=34​ (square units)

Q. The length of sub-normal at any point $P (x, y)$ on the curve, which is passing through $M (0, 1)$ is unity. The area bounded by the curves satisfying this condition is equal to

$\frac{1}{3}$

$\frac{2}{3}$

$\frac{4}{3}$

$\frac{8}{3}$