If the normal at P on the ellipse x2-a2 - y2-b2 - 1 cuts the major and minor axes in Q and R respectively then PQ-PR -

Equation of the normal at P(x_1, y_1) is a^2x/x_1 b^2y/y_1 = a^2 b^2 Normal cuts major axis at Q ( a^2 b^2/a^2 x_1, 0 ) and minor axis at (0, a^2+b^2/b^2 ...

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Byju's Answer

Standard XII

Mathematics

Point Form of Normal: Ellipse

If the normal...

Question

If the normal at P on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ cuts the major and minor axes in Q and R respectively then PQ:PR =

a : b

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b : a

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a^{2} : b^{2}

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b^{2} : a^{2}

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Solution

The correct option is D $b^{2} : a^{2}$
Equation of the normal at $P (x_{1}, y_{1})$ is $\frac{a^{2} x}{x_{1}} - \frac{b^{2} y}{y_{1}} = a^{2} - b^{2}$

Normal cuts major axis at $Q (\frac{a^{2} - b^{2}}{a^{2}} x_{1}, 0)$ and minor axis at $(0, \frac{- a^{2} + b^{2}}{b^{2}} y_{1})$
Now $P Q^{2} =$ ${(\frac{a^{2} - b^{2}}{a^{2}} x_{1} - x_{1})}_{1}^{2} + y_{1}^{2} = \frac{b^{4}}{a^{4}} x_{1}^{2} + y_{1}^{2} = \frac{b^{4} x_{1}^{2} + a^{4} y_{1}^{2}}{a^{4}}$
$\frac{P Q^{2}}{P R^{2}} = \frac{b^{4}}{a^{4}} \Rightarrow \frac{P Q}{P R} = \frac{b^{2}}{a^{2}}$

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Similar questions

Q. If the normal at P on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

cuts the major and minor axes in Q and R respectively then PQ:PR =

Q. If the normal at P on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

cuts the major and minor axes in Q and R respectively the PQ:PR =

Q. If the normal at a point

P

on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

of semi axes

a, b

& centre

C

cuts the major &minor axes at

G

g

, then

Q. If the normal at any point

P

on the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

meets the axes in

G

and

g

respectively, then

P G : P g =

Q. A normal inclined at an angle of

\frac{π}{4}

to the x-axis of the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

is drawn. It meets the major and minor axes in P and Q respectively. If C is the centre of the ellipse then the area of the triangle CPQ is

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If the normal at P on the ellipse x2a2+y2b2=1 cuts the major and minor axes in Q and R respectively then PQ:PR =

The correct option is D b2:a2Equation of the normal at P(x1,y1) is a2xx1−b2yy1=a2−b2 Normal cuts major axis at Q(a2−b2a2x1,0) and minor axis at (0,−a2+b2b2y1) Now PQ2= (a2−b2a2x1−x1)2+y21=b4a4x21+y21=b4x21+a4y21a4 PQ2PR2=b4a4⇒PQPR=b2a2

If the normal at P on the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ cuts the major and minor axes in Q and R respectively then PQ:PR =