find the length of the chord joining the points in which the straight line, (x/a)+(y/b)=1, meets the circle x^2+y^2=r^2

Question

Ashwin Muralidharan IIT Madras · Answer

Hi Suhas,

Consider (p,q) to be the midpoint of the chord.

So T=S1 is its equation.

which is px+qy = p²+q²---------(this eqn is the same as x/a+y/b=1)

So ap = bq = ??p²+q².

Let the line intersect the cirlcea at (x1,y1) and (x2,y2)

So D = √[ (x1-x2)²+(y1-y2)² ] = √[ (x1+x2)²+(y1+y2)² - 4x1x2 - 4y1y2 ]

Now x1+x2 = 2p, y1+y2 = 2q --------(As (p,q) is the midpoint).

From the line equation px+qy = p²+q², eliminate y from the circle eqn. You'll get a QE in x, for which x1,x2 are the roots.

So you'll get product of roots = x1x2 = p²+q² - r²q².

similarly y1y2 = p²+q² - r²p².

Substitute in the distance, and you'll get

D = √[4r² - 4(p²+q²)]

Now ap=bq=p²+q².

Also p/a+q/b=1

So p²+q² = (a²b²)/(a²+b²)......

Now substitute and get D, interms of r,a,b

Best Regards,

Ashwin (IIT Madras).

User · Answer

From the circle centre (0,0), drop a perpendicular to the line, bx+ay-ab=0. Its length, p=ab/√(a²+b²) The chord length, L = 2[√(r² -p²)] = 2√[(r² -(ab)²/(a²+b²))].

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find the length of the chord joining the points in which the straight line, (x/a)+(y/b)=1, meets the circle x^2+y^2=r^2

find the length of the chord joining the points in which the straight line, (x/a)+(y/b)=1, meets the circle x^2+y^2=r^2

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