Exercise · ZW-TANGENT-PLANE-009 · ★★★ Advanced
A tangent plane to \(z=x^2+y^2\) is parallel to the plane \(z=4x-2y+7\). What is the point of tangency \((a,b)\)?
Choose one option.
Show answer
- C. \((2,-1)\)
Explanation
Parallel planes have the same slopes: \(f_x=2a=4\) and \(f_y=2b=-2\), so \((a,b)=(2,-1)\).
Review the concept: Tangent plane
The tangent plane to the graph of a function \(f(x,y)\) at the point \((a,b,f(a,b))\) is the plane that best approximates the surface near that point and contains the tangent lines of its two coordinate sections.
\[z=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\]
More practice problems: tangent plane
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- ZW-TANGENT-PLANE-003 Find the tangent plane to z=x^2y at (1,1,1).
- ZW-TANGENT-PLANE-004 Find the tangent plane to z=√x^2+y^2 at the point (3,4,5).
- ZW-TANGENT-PLANE-005 The tangent plane to z=x^2+3y at the point (1,0,1) has normal vector:
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- ZW-TANGENT-PLANE-008 The tangent plane to z=x^2+y^2 at (1,1,2) meets the z-axis at z= …