the line x =2y intersects the ellipse x2/4 + y2 =1 at the point P and Q. the equation of the circle with PQ as diameter is
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Answer:
x² + y² = 20 / 17
Step-by-step explanation:
As the ellipse is centred at the origin and the line y = 2x passes through the origin, the circle will be centred at the origin.
So the equation of the circle will be
x² + y² = r²
where r is the radius, and this is the distance from the origin to P.
Let P = ( u, v ).
Since P is on the line, we have v = 2u.
Since P is on the ellipse, we have:
u²/4 + v² = 1 => u²/4 + (2u)² = 1 => u²/4 + 4u² = 1
=> u² + 16u² = 4 => 17u² = 4
=> u² = 4 / 17
Also, since r is the distance from the origin to P, we have
r² = u² + v² = u² + (2u)² = u² + 4u² = 5u² = 5 × 4 / 17 = 20 / 17.
Therefore the equation of the circle is
x² + y² = 20 / 17.
muqaddasishaq987:
thanks
but actually options of this ques are a. x2 + y2 = 1/2 b. x2 + y2 = 1 c. x2 + y2 = 2 d. x2 + y2 = 5/2
why u take u=2v
The line given by y=2x is "the line whose points have y-coordinate equal to 2 times their x-coordinate". The point P=(u,v) is on this line, so its y-coordinate (v) is 2 times its x-coordinate (u). That is, since P=(u,v) is on the line y=2x, we have v=2u.
Just rechecked solution for you. Also checked with GeoGebra (plot ellipse and line, get P, Q as points of intersection, draw circle on diameter PQ). The solution above is correct. It would seem that either there is a mistake in the question (i.e. it's not what was intended), or whoever gave the options got it wrong.
ok thanks!
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