The area of the triangle formed by the tangen | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the curve $y^2=4x$ and the point $P(1,2)$.The slope of the tangent at $(1,2)$ is found by differentiating $y^2=4x$: $2y \frac{dy}{dx} = 4 \R

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(D) Given the curve $y^2=4x$ and the point $P(1,2)$.The slope of the tangent at $(1,2)$ is found by differentiating $y^2=4x$: $2y \frac{dy}{dx} = 4 \Rightarrow \frac{dy}{dx} = \frac{2}{y}$. At $(1,2)$,the slope $m_t = \frac{2}{2} = 1$.The equation of the tangent is $y-2 = 1(x-1) \Rightarrow y = x+1$.The tangent intersects the $Y$-axis $(x=0)$ at $A(0,1)$.The slope of the normal $m_n = -\frac{1}{m_t} = -1$.The equation of the normal is $y-2 = -1(x-1) \Rightarrow y = -x+3$.The normal intersects the $Y$-axis $(x=0)$ at $B(0,3)$.The triangle is formed by vertices $P(1,2)$,$A(0,1)$,and $B(0,3)$.The base of the triangle along the $Y$-axis is the distance between $A(0,1)$ and $B(0,3)$,which is $|3-1| = 2$ units.The height of the triangle is the perpendicular distance from $P(1,2)$ to the $Y$-axis,which is the $x$-coordinate of $P$,i.e.,$1$ unit.Area $= \frac{1}{2} 	imes 	ext{base} 	imes 	ext{height} = \frac{1}{2} 	imes 2 	imes 1 = 1$ square unit.

The area of the triangle formed by the tangent and the normal drawn to the curve $y^2=4x$ at $(1,2)$ with the $Y$-axis is (in square units):

Similar Questions

The sum of the squares of the intercepts on the axes made by a tangent at any point on the curve $x^{2/3} + y^{2/3} = a^{2/3}$ is -

Find the points on the curve $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$ at which the tangents are parallel to the $x$-axis.

The points on the curve $y^2 = \frac{x^3}{9}$,where the normal to the curve makes equal intercepts with the axes,are

If $y=4x-5$ is a tangent to the curve $y^2=px^3+q$ at $(2,3)$,then the values of $p$ and $q$ are respectively

At which point on the curve $y^2 = x$ does the tangent make an angle of $45^{\circ}$ with the $x-$axis?

Vedclass Test Series

Exam Paper Generator

Online Exam Module