Let a tangent to the curve 9 x2+16 y2=144 intersect the coordinate axes at the points A and B. Then, the mininum length of the line segment AB is

Question

Let a tangent to the curve 9 x2+16 y2=144 intersect the coordinate axes at the points A and B. Then, the mininum length of the line segment AB is (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Equation of tangent at point P (4 cos θ, 3 sin θ) is ( x cos θ/4)+( y sin θ/3)=1 So A is (4 sec θ, 0) and point B is (0,3 operatornamecosec θ) Length A B =√16 sec 2 θ+9 operatornamecosec2 θ =√25+16 tan 2 θ+9 cot 2 θ ≥ 7

Q. Let a tangent to the curve $9 x^{2} + 16 y^{2} = 144$ intersect the coordinate axes at the points $A$ and $B$ . Then, the mininum length of the line segment $A B$ is

Equation of tangent at point $P (4 cos θ, 3 sin θ)$ is $\frac{x c o s θ}{4} + \frac{y s i n θ}{3} = 1$
So $A$ is $(4 sec θ, 0)$ and point $B$ is $(0, 3 cosec θ)$
Length $A B = 16 sec^{2} θ + 9 cosec^{2} θ$ $= 25 + 16 tan^{2} θ + 9 cot^{2} θ \geq 7$

Q. Let a tangent to the curve 9x2+16y2=144 intersect the coordinate axes at the points A and B. Then, the mininum length of the line segment AB is

Equation of tangent at point P(4cosθ,3sinθ) is 4xcosθ​+3ysinθ​=1 So A is (4secθ,0) and point B is (0,3cosecθ) Length AB=16sec2θ+9cosec2θ​ =25+16tan2θ+9cot2θ​≥7

Q. Let a tangent to the curve $9 x^{2} + 16 y^{2} = 144$ intersect the coordinate axes at the points $A$ and $B$ . Then, the mininum length of the line segment $A B$ is

Equation of tangent at point $P (4 cos θ, 3 sin θ)$ is $\frac{x c o s θ}{4} + \frac{y s i n θ}{3} = 1$
So $A$ is $(4 sec θ, 0)$ and point $B$ is $(0, 3 cosec θ)$
Length $A B = 16 sec^{2} θ + 9 cosec^{2} θ$ $= 25 + 16 tan^{2} θ + 9 cot^{2} θ \geq 7$