To find the tangent plane to the surface f(x,y,z)=0 at a point (X,Y,Z) we use the following method:
<span>Calculate grad f = (f_x, f_y, f_z). The normal vector to the surface at the point (X,Y,Z) is grad f(X,Y,Z). The equation of a plane with normal vector n which passes through the point p is (r-p).n=0, where r=(x,y,z) is the position vector. So the equation of the tangent plane to the surface through the point (X,Y,Z) is ((x,y,z)-(X,Y,Z)).grad f(X,Y,Z)=0. </span>
<span>Now in your case we have f(x,y,z)=y-x^2-z^2, so grad f=(-2x,1,-2z), and the equation of the tangent plane at the point (X,Y,Z) is </span>
<span>((x,y,z)-(X,Y,Z)).(-2X,1,-2Z)=0, </span>
<span>that is </span>
<span>-2X(x-X)+1(y-Y)-2Z(z-Z)=0, </span>
<span>i.e. </span>
<span>-2Xx+y-2Zz = -2X^2+Y-2Z^2. (1) </span>
<span>Now compare this equation with the plane </span>
<span>x + 2y + 3z = 1. (2) </span>
<span>The two planes a_1x+b_1y+c_1z=d_1, a_2x+b_2y+c_2z=d_2 are parallel when (a_1,b_1,c_1) is a multiple of (a_2,b_2,c_2). So the two planes (1),(2) are parallel when (-2X,1,-2Z) is a multiple of (1,2,3), and we have </span>
<span>(-2X,1,-2Z)=1/2(1,2,3) </span>
<span>for X=-1/4 and Z=-3/4. On the paraboloid the corresponding y coordinate is Y=X^2+Z^2=1^4+9^4=5/2. </span>
<span>So the tangent plane to the given paraboloid at the point (-1/4,5/2,-3/4) is parallel to the given plane.</span>
In fact, when the lights are connected in parallel, they are connected on separate branches to the source of voltage, so if one light bulb burns out, the other lights continue to work because the current continues to flow in the other branches of the circuit.
Vice-versa, if the light bulbs are connected in series, they are on the same branch This means that if one of them burns out, the circuit is open in that point, so the current cannot flow anymore and the other light bulbs turn off as well.
We shall consider two properties: 1. Temperature difference 2. Thermal conductivity of the material
Use a cylindrical rod of a given material (say steel) which is insulated around its circumference.
One end of the rod is dipped in a large reservoir of water at 100 deg.C and the other end is dipped in water (with known volume) at 40 deg. C. The cold water if stored in a cylinder which is insulated on all sides. A thermometer reads the temperature of the cold water as a function of time.
This experiment will show that (a) heat flows from a region of high temperature to a region of lower temperature. (b) The thermal energy of a body increases when heat is added to it, and its temperature will rise. (c) The thermal conductivity of water determines how quickly its temperature will rise. If mercury replaces water in the cold cylinder, its temperature will rise at a different rate because its thermal conductivity is different.
<span>So the question is what gases form the atmosphere of Uranus. So Uranus and Neptune are classified as "ice giants". They have a similar atmosphere to "gas giants" Saturn and jupiter and Neptunes atmosphere is primarily composed out of hidrogen and helyum. So the correct answer is a.</span>
Distance is total path travelled by an object during its entire journey. It is a scalar quantity i.e only magnitude.
Displacement is the shortest distance covered by an object. It is basically the change in position of object. It is a vector quantity i.e direction as well as magnitude.
When an object is travelling in a straight line and stops at the end point, then both distance and displacement are same.
When an object is travelling in a straight line and then changes its direction or we can say come backwards then the magnitude of distance and displacement are different.
