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NTA Abhyas 2020

Exam: NTA Abhyas
Year: 2020
Subject: Mathematics
Chapter: Ellipse

NTA Abhyas 2020 · Mathematics · Ellipse

If any tangent to the ellipse 25x^ 2 + 9y^ 2 = 225 meets the...

If any tangent to the ellipse

25x^{2} + 9y^{2} = 225

meets the coordinate axes at

A

and

B

such that

OA = OB

then, the length

AB

is equal to (where,

O

is the origin)

Options

A $\sqrt{17}$ units
B $\sqrt{34}$ units
C $2\sqrt{17}$ units
D $2\sqrt{34}$ units

Correct answer

C. $2\sqrt{17}$ units

Step-by-step solution

Given, equation of the ellipse is

\dfrac{{x}^{2}}{9} + \dfrac{{y}^{2}}{25} = 1

Let, a point on it be

P\left(3cos \theta,5sin \theta\right)

and equation of the tangent at

P

is

\dfrac{x}{3}cos \theta + \dfrac{y}{5}sin \theta = 1

which meets the

x

-axis at

A\left(\dfrac{3}{cos\theta}, 0\right)

and

y

-axis at

B\left(0, \dfrac{5}{sin\theta}\right)

Let,

OA = OB = \lambda

cos \theta = \dfrac{3}{\lambda}, sin \theta = \dfrac{5}{\lambda}

Now,

co{s}^{2}\theta + si{n}^{2}\theta = 1

\dfrac{9}{{\lambda}^{2}} + \dfrac{25}{{\lambda}^{2}} = 1 ⇒ {\lambda}^{2} = 34

∴ \lambda = \sqrt{34}

Now,

AB = \sqrt{2}\lambda = 2\sqrt{17}units

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