Finding Tangent plane: IIT JAM 2018 problem 5

[et_pb_section fb_built="1" _builder_version="3.22.4"][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

What are we learning?

[/et_pb_text][et_pb_text _builder_version="4.0.9" text_font="Raleway||||||||" text_font_size="18px" background_color="#f4f4f4" custom_margin="50px||50px||false|false" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]We will learn to find tangent plane by solving an IIT JAM 2018 Problem. This is the Question no. 5 of the IIT JAM 2018 Solved Paper Series. Go through this link for Question no. 6. Gradient is one of the key concepts of vector calculus. We will use this problem from IIT JAM 2018 to clear our concepts.

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Understand the problem

[/et_pb_text][et_pb_text _builder_version="4.0.9" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="50px||50px||false|false" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]The tangent plane to the surface $z= \sqrt{x^2+3y^2}$ at (1,1,2) is given by

$x-3y+z=0$
$x+3y-2z=0$
$2x+4y-3z=0$
$3x-7y+2z=0$

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.0.9" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.0.9"]IIT Jam 2018[/et_pb_accordion_item][et_pb_accordion_item title="Key competency" _builder_version="4.0.9" open="off"]Gradient[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.0.9" open="off"]Easy[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.0.9" open="off"]

Calculus: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability – Vol 2

Tom M. Apostol

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Look at the knowledge graph...

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Start with hints

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="4.0.9" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.0.9"]Given a differentiable function $Z=f(x,y)$ , Observe that when we are asked to find a tangent plane at $(x_0,y_0,z_0)$ then the picture that comes in our mind is a plane that touches the curve at a point.

When we are in dimension $2$ it is just a line, (easy to visualize), dim 3 a plane (still visible), dim 4,5,…. a surface which is hard to see, but we can plug in $x=x_0$ in the equation $z=f(x,y)$ to have $z=f(x_0,y)$ which is just a curve in 2D then we can visualize the tangent line at $y=y_0$ is a part of the tangent plane $z=f(x,y)$ isn’t it?? The same thing is true of about the tangent line at $x=x_0$ for the curve $z=f(x,y_0)$ . These $f(x,y_0)$ and $f(x_0,y)$ are called sections of the curve $f(x,y)=z$ . Here $(x_0,y_0,z_0)=(1,2,3)$ . So, quickly find out $f(1,y)$ and $f(x,1)$ .

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.0.9"]You can see that $f(1,y)=\sqrt{1+3y^{2}}$ and $f(x,1)= \sqrt{x^{2}+3}$ Now observe that the tangent plane of the curve $z=f(x,y)$ is a plane right !! What will be the basic structure of a plane at $(x_0,y_0,z_0)$ ?

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.0.9"]

It is a $a(x-x_0)+ b(y-y_0)+ c(z-z_0)=0$ ----------------------(1) Now see that $(x_0,y_0,z_0)=(1,1,2)$ is already given in the question. Hence the unknown is $(a,b,c)$ . Equation (1) implies $z = z_0+ \frac{a}{c}(x-x_0)+ \frac{b}{c}(y-y_0)$ Differentiating the equation by $x$ we get, $z_x= \frac{a}{c}$ Differentiating the equation by $y$ we get, $z_y= \frac{b}{c}$ Hence the equation of the tangent plane is $z=z_0+z_x|_{(x_0,y_0)}(x-x_0)+ z_y|_{(x_0,y_0)}(y-y_0)$ So calculate $z_x$ and $z_y$ at $(x_0,y_0)$

[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.0.9"]

$z_x = \frac{d}{dx}f(x,1)= \frac{2x}{2\sqrt{x^{2}+3}}|_{(1,1)} = \frac{2}{4}= \frac{1}{2}$ $z_y=\frac{d}{dy}f(1,y)=\frac{6y}{2\sqrt{1+3y^2}}|_{(1,1)}=\frac{6}{2 \times 2}=\frac{3}{2}$ So the equation of the tangent line is $z= 2+\frac{1}{2}(x-1)+\frac{3}{2}(y-1)$ $\Rightarrow 2z= 4+x-1+3y-3$

$x+3y-2z=0$ (Ans)

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Try to answer this question

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Play with graph

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Connected Program at Cheenta

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The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.[/et_pb_blurb][et_pb_button button_url="https://www.cheenta.com/collegeprogram/" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" background_layout="dark" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3"][/et_pb_button][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

What are we learning?

[/et_pb_text][et_pb_text _builder_version="4.0.9" text_font="Raleway||||||||" text_font_size="18px" background_color="#f4f4f4" custom_margin="50px||50px||false|false" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]We will learn to find tangent plane by solving an IIT JAM 2018 Problem. This is the Question no. 5 of the IIT JAM 2018 Solved Paper Series. Go through this link for Question no. 6. Gradient is one of the key concepts of vector calculus. We will use this problem from IIT JAM 2018 to clear our concepts.

[/et_pb_text][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Understand the problem

[/et_pb_text][et_pb_text _builder_version="4.0.9" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="50px||50px||false|false" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]The tangent plane to the surface $z= \sqrt{x^2+3y^2}$ at (1,1,2) is given by

$x-3y+z=0$
$x+3y-2z=0$
$2x+4y-3z=0$
$3x-7y+2z=0$