The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight - Maths - Mathematical Reasoning

Question

The locus of the
mid-point of the chord of  contact of tangents drawn from points lying on the straight
line

4x 5y = 20 to the circle x 2 + y 2 = 9 is
(A) 20(x 2 + y 2 ) 36x + 45y =
0 (B) 20(x 2 + y 2 ) + 36x 45y = 0
(C) 36(x 2 + y
2 ) 20x + 45y = 0 (D) 36(x
2 + y 2
) + 20x 45y = 0

Ajanta Trivedi · Accepted Answer

let the point on the line 4x-5y=20 be A let the coordinates of the point A is given by $(a, \frac{4 a - 20}{5})$ then the equation of the chord of contact from point A to the circle $x^{2} + y^{2} = 9$ is given by; $a x + (\frac{4 a - 20}{5}) y = 9 5 a x + (4 a - 20) y = 45 (1)$ let the locus of the mid-point of the chord of contact be P(h,k) therefore the equation of the chord with mid-point (h,k) is given by; $h x + k y = h^{2} + k^{2} (2)$ equation (1) and (2) represent the same straight line, therefore $\frac{5 a}{h} = \frac{4 a - 20}{k} = \frac{45}{h^{2} + k^{2}}$ (3) $\frac{5 a}{h} = \frac{4 a - 20}{k} 5 a k = 4 a h - 20 h a (5 k - 4 h) = - 20 h a = \frac{20 h}{4 h - 5 k}$ now substitute the value of a, $\frac{5 a}{h} = \frac{45}{h^{2} + k^{2}} \frac{5}{h} \cdot \frac{20 h}{4 h - 5 k} = \frac{45}{h^{2} + k^{2}} \frac{20}{4 h - 5 k} = \frac{9}{h^{2} + k^{2}} 20 (h^{2} + k^{2}) = 36 h - 45 k 20 (h^{2} + k^{2}) - 36 h + 45 k = 0$ since P(h,k) is a variable point therefore replace h by x and k by y the required locus is $20 (x^{2} + y^{2}) - 36 x + 45 y = 0$

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x 5y = 20 to the circle x 2 + y 2 = 9 is (A) 20(x 2 + y 2 ) 36x + 45y = 0 (B) 20(x 2 + y 2 ) + 36x 45y = 0 (C) 36(x 2 + y 2 ) 20x + 45y = 0 (D) 36(x 2 + y 2 ) + 20x 45y = 0

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line

4x 5y = 20 to the circle x² + y² = 9 is

(A) 20(x² + y²) 36x + 45y = 0 (B) 20(x² + y²) + 36x 45y = 0

(C) 36(x² + y²) 20x + 45y = 0 (D) 36(x² + y²) + 20x 45y = 0