Question

The pole of the straight line \[x+4y=4\] with respect to ellipse \[{{x}^{2}}+4{{y}^{2}}=4\] is:
(a) \[(1,4)\]
(b) \[(1,1)\]
(c) \[(4,1)\]
(d) \[(4,4)\]

Answer 1

Hint: Assume a point as a pole, then find the polar equation of the given curve and compare it with \[x+4y=4\]. The polar is given by \[{{S}_{1}}=0\].

Complete step-by-step answer:

Given that we need to find the pole of a given polar with respect to the ellipse.

Let the pole be assumed as\[({{x}_{1}},{{y}_{1}})\].

Then the polar of \[({{x}_{1}},{{y}_{1}})\]with respect to \[{{x}^{2}}+4{{y}^{2}}=4\] is given as:

\[x{{x}_{1}}+4y{{y}_{1}}=4\]….(1)

Since, the polar for any curve \[S=0\] is given as \[{{S}_{1}}=0\].

The polar has already been mentioned in the question as:

\[x+4y=4\]...(2)

Now by carefully comparing the equations of (1) and (2), we will have:

Coefficient of \[x\] in equation (1) = coefficient of \[x\] in equation (2)

\[{{x}_{1}}=1\]

Similarly, coefficient of \[y\] in equation(1) = coefficient of \[y\] in equation (2)

\[4{{y}_{1}}=4\]

\[{{y}_{1}}=1\]

Therefore, the pole \[({{x}_{1}},{{y}_{1}})\] is \[(1,1)\].


NOTE: Pole is a point and polar is a straight line for a given plane, do not get confused.
Students often make mistakes when finding the polar. Polar of a curve \[S=0\] is given by \[{{S}_{1}}=0\].

They sometimes make mistakes when comparing the general equation of the polar with the given equation of the polar.