If the locus of middle points of portions of tangents intercepted between the axes of the ellipse a 2 x 2 + b 2 y 2 = 1 is x 2 a 2 + y 2 b 2 = λ , then the numerical value of λ is

Tangent at ' α ' is a x cos α + b y sin α = 1 Let A ( a sec α , 0 ) B ( 0 , b cos ec α ) ( h , k ) is mid point h = 2 1 a sec α k = 2 b cos ec α ⇒ 4 x 2 a 2 + 4 y 2 b 2 = 1

Mathematics - Ellipse Question with Solution | TestHub

MathematicsEllipseTangent to ellipseMedium2 minQB

MathematicsMediuminteger

If the locus of middle points of portions of tangents intercepted between the axes of the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ is $\frac{a ^{2}}{x ^{2}} + \frac{b ^{2}}{y ^{2}} = λ$ , then the numerical value of $λ$ is

Answer:

Solution:

Tangent at ' $α$ ' is $\frac{x}{a} cos α + \frac{y}{b} sin α = 1$

Let $A (a sec α, 0) B (0, b cos ec α)$

$(h, k)$ is mid point

$h = \frac{1}{2} a sec α k = \frac{b}{2} cos ec α$ $\Rightarrow \frac{a ^{2}}{4 x ^{2}} + \frac{b ^{2}}{4 y ^{2}} = 1$

Stream:JEESubject:MathematicsTopic:EllipseSubtopic:Tangent to ellipse

⏱ 2mℹ️ Source: QB

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