From a point P tangents are drawn to the ellipse x2/a2 +y2/b2 = 1 . if the chord of contact touches the ellipse x2/c2 +y2/d2 = 1 then locus of P is?

We are given two ellipses:

Ellipse 1:
x²/a² + y²/b² = 1

Ellipse 2:
x²/c² + y²/d² = 1

Step 1: Equation of Chord of Contact
If (h, k) is an external point, then the equation of the chord of contact of tangents drawn from (h, k) to the first ellipse is given by

(hx/a²) + (ky/b²) = 1

This is the equation of the chord of contact.

Step 2: Condition for Tangency
This chord of contact is also a tangent to the second ellipse. The condition for tangency of a line Ax + By = C to the ellipse x²/c² + y²/d² = 1 is

A²/c² + B²/d² = C²

Substituting A = h/a², B = k/b², and C = 1, we get

(h²/a⁴)/c² + (k²/b⁴)/d² = 1²

h² / (a⁴c²) + k² / (b⁴d²) = 1

Step 3: Locus of P
Replacing (h, k) by (x, y), the required locus of P is

x² / (a⁴c²) + y² / (b⁴d²) = 1

This represents an ellipse.