Question

Let $y=f\left(x\right)$ and $y=g\left(x\right)$ be the pair of curves such that
(i) The tangents at point with equal abscissae intersect on y-axis.
(ii) The normal drawn at points with equal abscissae intersect on x-axis and
(iii) curve f(x) passes through $(1,1)$ and $g\left(x\right)$ passes through $(2,3)$ then the value of ${\int }_1^2\left(g\right(x)-f(x\left)\right)dx$ is?

• A. $2$
• B. $3$
• C. $4$
• D. $5$

Answer 1

The correct option is A $2$

$y=f\left(x\right)$ and $y=g\left(x\right)$

(i)

$\frac{dy}{dy}=c\Rightarrow c_1=1$

for second curve

$\frac{dy}{dy}=c_2\Rightarrow c_2=1$

(ii)

$\frac{dy}{dx}=d_1---\left(1\right)$

for second curve

$\frac{dy}{dx}=d_2----\left(2\right)$

(iii)

From eq (1) and (2)

$d_1=2=d_2$

${\int }_1^2\left(g\right(x)-f(x\left)\right)d\left(x\right)$

${\int }_1^2(4-2)d\left(x\right)$

$2{\int }_1^{2}d\left(x\right)$

$2(2-1)=2$