Question:
The subtangent, ordinate and subnormal to the parabola $y^2= 4ax$ at a point (different from the Origin) are in
The subtangent, ordinate and subnormal to the parabola $y^2= 4ax$ at a point (different from the Origin) are in
Updated On: Aug 1, 2022
- G.P
- A.P.
- H.P
- none of these
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The Correct Option is A
Solution and Explanation
$y^{2} = 4ax$
$\Rightarrow 2\,y \frac{dy}{dx} = 4a$
$\Rightarrow \frac{dy}{dx} = \frac{2a}{y}$
Subtangent $= \frac{y}{dy / dx}=\frac{y}{2\, a/y}$
$= \frac{y^{2}}{2a} = \frac{4ax}{2a}= 2x$
Ordinate $= y = \sqrt{4\,ax} = 2\sqrt{a} \sqrt{x}$
Subnormal $= y \frac{dy}{dx} = 2a$
Since (subtangent) (subnormal) $= \left(2x\right)\left(2a\right)$
$= 4ax =$ (ordinate)$^{2}$
$\Rightarrow$ subtangent, ordinate and subnormal are in $G.P$.
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Notes on Tangents and Normals
Concepts Used:
Tangents and Normals
- A tangent at a degree on the curve could be a straight line that touches the curve at that time and whose slope is up to the derivative of the curve at that point. From the definition, you'll be able to deduce the way to realize the equation of the tangent to the curve at any point.
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- A normal at a degree on the curve is a line that intersects the curve at that time and is perpendicular to the tangent at that point. If its slope is given by n, and also the slope of the tangent at that point or the value of the derivative at that point is given by m. then we got
m×n = -1
- The normal to a given curve y = f(x) at a point x = x0
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Diagram Explaining Tangents and Normal:
