The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x - y = 20 to the circle x² + y² = 9 is

Given: 4x - y = 20
Let midpoint of the chord of contact be (h, k). Then Equation of the chord is with the above midpoint will be given as hx + ky = h² + k² (1) (using the general equation of midpoint chord relation)
Let any point on the line 4x - y = 20 be P(α, 4α - 20). Then using the relation of the equation of chord of contact from the given outside point P we get, αx + (4α - 20)y = 9 (2)
Since (1) and (2) represent the same equation of chord, equating the coefficients
hα = k
(4α - 20) = 9k/(h² + k²)
α = k/h
4α - 20 = 9k/(h² + k²)
Substitute α = k/h in 4α - 20 = 9k/(h² + k²)
4(k/h) - 20 = 9k/(h² + k²)
4k - 20h = 9kh/(h² + k²)
(4k - 20h)(h² + k²) = 9kh
4kh² + 4k³ - 20h³ - 20hk² = 9kh
4k³ + 4kh² - 20h³ - 20hk² - 9kh = 0
Multiply by 5
20k³ + 20kh² -100h³ -100hk² -45kh = 0
20k³ + 20kh² - 100h³ - 100hk² - 45kh = 0
Assuming it to be a circle with center at origin, we assume the equation to be of the form 20(h²+k²) + ah + bk = 0. Comparing this with the obtained equation, it will not be possible to obtain the equation in this form.
Let's proceed differently.
α = 9h/(h² + k²) (3)
Also 4α - 20 = 9k/(h² + k²) ⇒ 4α = 9k/(h² + k²) + 20 ⇒ α = [9k + 20(h² + k²)]/[4(h² + k²)] (4)
Eliminate 'α' from '3' and '4' to get the locus.
4 × 9h = 45k + 20(h² + k²)
36h = 45k + 20(h² + k²)
20(h² + k²) - 36h + 45k = 0
∴ Locus of the midpoint is 20(x² + y²) - 36x + 45y = 0