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Grade 11Algebra

the feet of perpendicular from the origin on a variable chord of the circle X² + Y² -2 X - 2y = 0 is N. if the variable chord makes an angle of 90 degree at the origin then the locus of N has the equation....?

Janhavi

7 Years agoGrade 11

2 Answers

Dr Bhishma Hazarika

7 Years ago

Let (a,b) is the foot of the perpendicular drawn on the variable chord of the circle from the origin.

So, Equation of the chord $y-b=-a/b(x-a)$

or $ax+by={a}^2 +b^2$

or $\frac{ax+by}{a^2+b^2 }=1$ ….....................(1)

Now hoogeneous form of the given circle by equation is $x^2+y^2-2x\frac{ax+by}{a^2+b^2}-2y\frac{ax+by}{a^2+b^2}=0$

It will represent pair of st.lines. but by question chord makes 90⁰ at the origin;

so co-officiant of $x^2$ +co-officiant of $y^2$ is zero

i.e

Dr Bhishma Hazarika

7 Years ago

Continue from the first answer......

$(1-\frac{2a}{a^2+b^2})+(1-\frac{2b}{a^2+b^2})= 2(a^2+b^2)-2a-2b=0$

so locus of the foot of the perpendicular from the orogin is another circle

$2(x^2+y^2)-2x-2y=0$

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