a Circle C1 is drawn having any point P on x axis as its centre and - Maths - Circles

Question

a Circle C1 is drawn having any point P on x axis as its centre and
passing through the centre of the circle C:
x2+y2=1 A common tangent to C1,
and C intersects the circle at Q and R respectively Then Q(x,y)
always satisfies x2 = lamda, find the value of lamda

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

Let the point be Ph,0
 Equation of circle is x-h2+y2=r2It passes through centre of x2+y2=1 which is 0,0, hence0-h2+0=r2⇒h2=r2Hence equation will bex-h2+y2=h2
iA tangent to a circle is perpendicular to the radius at the point of tangency, so an equation of the tangent at Q isxQ-hx+yQy=xQ-hxQ+y2Q=xQ2-hxQ+y2Q=xQ2+h2-2hxQ+hxQ-h2+y2Q=xQ-h2+yQ2+hxQ-h=h2 +hxQ-h2 Using ixQ-hx+yQy=hxQ ⇒xQ-hx+yQy-hxQ=0Since this line is also tangent to C, its distance  from the origin must be equal to the radius of C
hxQxQ-h2+yQ2=1 Radius of x2+y2=1 is 1 unit⇒hxQh2=1Square both sidesh2xQ2h2=1⇒xQ2=1⇒λ=1

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a Circle C1 is drawn having any point P on x axis as its centre and passing through the centre of the circle C: x2+y2=1. A common tangent to C1, and C intersects the circle at Q and R respectively. Then Q(x,y) always satisfies x2 = lamda, find the value of lamda.

a Circle C1 is drawn having any point P on x axis as its centre and passing through the centre of the circle C: x²+y²=1. A common tangent to C₁, and C intersects the circle at Q and R respectively. Then Q(x,y) always satisfies x²= lamda, find the value of lamda.