GATE2014 AE: GA-4

Question

If $y=5x^2+3$, then the tangent at $x=0$, $y=3$

• A. passes through $x=0,y=0$
• B. has a slope of $+1$
• C. is parallel to the $x$-axis
• D. has a slope of $-1$

Comments

To find the properties of the tangent line to the curve $y = 5x^2 + 3$ at the point $(0, 3)$, we need to determine its slope.

1. Find the Derivative

The slope of the tangent line at any point on a curve is given by the derivative of the function. Using the power rule for differentiation:

$$\frac{dy}{dx} = \frac{d}{dx}(5x^2 + 3)$$

$$\frac{dy}{dx} = 10x$$

2. Calculate the Slope at $x = 0$

Now, substitute the given $x$-coordinate ($x = 0$) into the derivative to find the specific slope at that point:

$$m = 10(0) = 0$$

A slope of 0 indicates that the tangent line is a horizontal line.

3. Analyze the Options

• A. passes through $x=0, y=0$: Since the tangent is horizontal and passes through $(0, 3)$, its equation is $y = 3$. This line never passes through the origin $(0, 0)$.

• B. has a slope of +1: Incorrect; the slope is 0.

• C. is parallel to the $x$-axis: Correct. All horizontal lines (slope = 0) are parallel to the $x$-axis.

• D. has a slope of -1: Incorrect; the slope is 0.

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Correct Answer:

C. is parallel to the $x$-axis

Answer 1

Given, $y=5x^2+3$
Slope $dy/dx = 10x$
Given point $x=0,y=3: (0,3)$
Now, at $(0,3)$ we have the slope as $dy/dx = 10 \times 0=0,$ which means no matter how much ever we change 'x',change in y is zero or y is a constant (y=3,here) i.e line is parallel to $X$- axis.
Options B and D eliminated as they have slope as '+1' and '-1'.
Option A is eliminated as given the tangent is at point $(0,3)$ and we found $slope=0$, which means it is parallel to $X$ axis. The equation of line is nothing but $y=3$, which never passes through $(0,0).$ Hence A too eliminated.
Answer: Option $C)$

Comments

https://www.khanacademy.org/math/algebra/two-var-linear-equations/hor-and-ver-lines-alg1/v/examples-of-slopes-and-equations-of-horizontal-and-vertical-lines

 

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If you can draw some common graphs like $x^2$, then almost no calculation is required in this question.

 

$x^2 \xrightarrow{\text{narrow down the graph in the x-direction 5 times}} 5x^2 \xrightarrow{\text{move the graph up in the y-direction till y = 3}} 5x^2 + 3$

at $(0, 3)$ graph $5x^2 + 3$ has global minima. so, tangent $y = 3$ will be parallel to $x$-axis. 

Answer 2

Option (c) 

Another approach :