mandeey wrote:

In the xy-plane, a circle C is drawn with center at (1, 2) and radius equal to 5. Is line l a tangent to the circle C?

(1) Point A with coordinates (a, b) lies on line l such that a(a-2)+ b(b-4) ≤ 20.

(2) The x-intercept of line l is 10.

In the xy-plane, a circle C is drawn with center at (1, 2) and radius equal to 5

from this information , we can find out the equation of circle .

\((x-1)^2 +(y-2)^2 = 5^2\)

Solving it , we get \(x^2 + 1 -2x +y^2 +4 - 4y = 25\)

\(x^2+y^2-2x-4y-20=0\)

Now we have to give yes/no answer to question: Is line l a tangent to the circle C?

Statement 1:Point A with coordinates (a, b) lies on line l such that a(a-2)+ b(b-4) ≤ 20.

Instead taking coordinate (a,b), we will cordinate as (x.y) in order to findout general equation.

We get \(x^2 +y^2 - 2x- 4y ≤ 20\)

Substracting 20 from both side,

\(x^2 +y^2 - 2x- 4y-20≤0\)

This implies that (x,y) or (a,b) according to statement 1 should lie on circumference of circle or inside it.

As Statement 1 gives us condition that point(a,b) of line l lies inside or on circumference of circle, it is insufficient for defining a particular line.

Statement 2: The x-intercept of line l is 10.

It means that line passes through point (10,0)

There can be infinite line passing through the point (10,0)

So this statement is also insufficient.

Combining Statement 1 and Statement 2 , Statemen1 gives condition for (a,b) on line l.

All Coordinate inside circle and on circumference will satisfy the condition.

There will be infinite number of line connecting point (a,b) and (10,0) , point as given by statement 2.

We do not have value of slope, in order to define a single line.

So Statement 1 and 2 together also are not sufficient.

So answer is E

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