the circle x^2+y^2=a^2 cuts off intercept on the straight line lx+my=1 which subtends an angle 45 at the origin.show that a^2(l^2+m^2)=4-2*1.414
Let be the intersect points between the line and the circle. Then .
The line AO has the slope and the line BO has the slope
The tangent of the acute angle between the two lines is
Replace and :
, (1)
The coordinates of A and B are the solutions of the system formed by the equation of the line and equation of the circle.
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Replace y in the equation of the circle and we have:
are the roots of the quadratic and we have
Replace in (1):
.
Now replace and .
Let .
Then
The positive root is .
.
Square both members:
The circle has radius and is centred at the origin. If a chord of the circle subtends an angle at the centre of the circle then the midpoint of the chord is at a distance from the origin. But , as you can check from the formula . It follows that .
The line intersects the circle at two points , whose x-coordinates are the roots of the quadratic equation , or . The sum of the roots of this equation is . Thus the midpoint of the chord has x-coordinate . Similarly, the midpoint of the chord has y-coordinate . Thus the distance of the midpoint from the origin is given by
Compare the two expressions for to see that .