Question

Let P be any point on the circle x² + y² = 25. Let L be the chord of contact of P with respect to the circle x^2 + y^2 = 9. The locus of the poles of the lines L with respect to the circle x² + y² = 36 is:

• A. \(y^2 = 20x\)
• B. \(\frac{x^2}{9} + \frac{y^2}{36} = 1\)
• C. \(x^2 + y^2 = 400\)
• D. \(\frac{x^2}{25} - \frac{y^2}{16} = 1\)

Hint:

Remember the equations for the chord of contact and polar of a point with respect to a circle.

Answer 1

The Correct Option is C

Step 1: Let P be a point on x² + y² = 25. Let P be (x₁, y₁). Since P lies on x² + y² = 25, we have x₁² + y₁² = 25.

Step 2: Find the chord of contact L of P with respect to x² + y² = 9.
The equation of the chord of contact L is given by xx₁ + yy₁ = 9.

Step 3: Find the pole of the line L with respect to x² + y² = 36.
Let the pole be (h, k).
The equation of the polar of (h, k) with respect to x² + y² = 36 is hx + ky = 36.
This must be the same as the chord of contact L: xx₁ + yy₁ = 9.
Comparing coefficients, we have:
\(\frac{h}{x_1} = \frac{k}{y_1} = \frac{36}{9} = 4\)
Thus, h = 4x₁ and k = 4y₁.
So, x₁ = \(\frac{h}{4}\) and y₁ = \(\frac{k}{4}\).

Step 4: Find the locus of the pole (h, k).
Since x₁² + y₁² = 25, we substitute x₁ = \(\frac{h}{4}\) and y₁ = \(\frac{k}{4}\):
\(\left(\frac{h}{4}\right)^2 + \left(\frac{k}{4}\right)^2 = 25\)
\(\frac{h^2}{16} + \frac{k^2}{16} = 25\)
h² + k² = 25 \(\times\) 16 = 400
Thus, the locus of the pole (h, k) is x² + y² = 400.
Therefore, the locus of the poles of the lines L with respect to the circle x² + y² = 36 is x² + y² = 400.