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JEE Main
Exam
JEE Main
Subject
Mathematics
Chapter
Ellipse

JEE Main · Mathematics · Ellipse

If P ( x _ 1 , y _ 1 ) is a point on the ellipse b^ 2 x^ 2 + a^ 2...

If P(x1,y1)isapointontheellipseP \left({x}_{1}, {y}_{1}\right) is a point on the ellipseb^{2}x^{2} + a^{2}y^{2} = a^{2}b^{2}thentheareaofΔthen the area of \DeltaSPS′ =If P ( x _ 1 , y _ 1 ) is a point on the ellipse b^ 2 x^ 2 + a^ 2 y^ 2 = a^ 2 b^ 2 then the area of Δ SPS′ =

Options

  1. Aaea2−x12ae\sqrt{a^{2} - {x}_{1}^{2}}
  2. Bbeb2−x12be\sqrt{b^{2} - {x}_{1}^{2}}
  3. Caeb2−x12ae\sqrt{b^{2} - {x}_{1}^{2}}
  4. Dbea2−x12be\sqrt{a^{2} - {x}_{1}^{2}}

Correct answer

D. bea2−x12be\sqrt{a^{2} - {x}_{1}^{2}}

Step-by-step solution


If P ( x _ 1 , y _ 1 ) is a point on the ellipse b^ 2 x^ 2 + a^ 2 y^ 2 = a^ 2 b^ 2 then the area of Δ SPS′ =

x2a2+y2b2=1⇒y12b2=1−x12a2\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 ⇒ \dfrac{{{y}_{1}}^{2}}{b^{2}} = 1 - \dfrac{{{x}_{1}}^{2}}{a^{2}}

y1=baa2−x12{y}_{1} = \dfrac{b}{a}\sqrt{a^{2} - {{x}_{1}}^{2}}

\Delta SPS′ = \dfrac{1}{2} \times 2ae \times {y}_{1} = ae \times \dfrac{b}{a}\sqrt{a^{2} - {{x}_{1}}^{2}}

\Delta SPS′ = be\sqrt{\left(a^{2} - {{x}_{1}}^{2}\right)}
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