Match the elements of Column I to elements of Column II....

Match the elements of Column I to elements of Column II. There can be single or multiple matches. Column I (a) The circles $x^{2}+y^{2}+6 x+k=0$ and $x^{2}+y^{2}+8 y-20=0$ are touching each other, if $\mathrm{k}$ equals (b) Shortest distance from $(2,-7)$ to the circle $x^{2}+y^{2}-14 x-10 y-151=0$ is (c) If the two circles $x^{2}+y^{2}+12 x+10 y+45=0$ and $x^{2}+y^{2}+2 x-14 y+a=0$ are touching each other externally, then the length of direct common tangent of the circles is (d) The circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-4 \lambda x+9=0$ have exactly two common tangents if $\lambda$ equals Column II (p) 8 (q) 2 (r) 12 (s) $-112$

Match the elements of Column I to elements of Column II. There can be single or multiple matches.
Column I
(a) The circles $x^{2}+y^{2}+6 x+k=0$ and $x^{2}+y^{2}+8 y-20=0$ are touching each other, if $\mathrm{k}$ equals
(b) Shortest distance from $(2,-7)$ to the circle $x^{2}+y^{2}-14 x-10 y-151=0$ is
(c) If the two circles $x^{2}+y^{2}+12 x+10 y+45=0$ and $x^{2}+y^{2}+2 x-14 y+a=0$ are touching each other externally, then the length of direct common tangent of the circles is
(d) The circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-4 \lambda x+9=0$ have exactly two common tangents if $\lambda$ equals
Column II
(p) 8
(q) 2
(r) 12
(s) $-112$

Key Concepts

Distance Formula

The distance formula, d = ?[(x? - x?)² + (y? - y?)²], is a fundamental tool used to calculate the straight-line distance between two points in the coordinate plane. It is particularly important in circle geometry for determining the distance between centers of circles, which in turn is crucial for establishing relationships such as tangency and intersection.

Standard Equation of a Circle

The standard equation of a circle in the coordinate plane is written as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form is obtained by completing the square from the general quadratic form and is essential for easily identifying a circle's geometric properties, such as its center and radius.

Tangency Condition

Tangency between two circles occurs when they touch each other exactly at one point. For two circles, this happens when the distance between their centers is equal to either the sum (for external tangency) or the absolute difference (for internal tangency) of their radii. This condition helps in setting up equations that involve unknown parameters in circle problems.

Distance from a Point to a Circle

The shortest distance from a point to a circle is found by first calculating the distance between the point and the center of the circle and then subtracting the radius of the circle. This concept is useful in determining how a point is positioned relative to a circle and is applicable in optimization problems and in determining tangents from external points.

Direct Common Tangents

Direct common tangents are lines that touch two circles externally without crossing the line connecting their centers. The length of a direct common tangent between two circles can be determined using the distance between their centers and their radii. This concept helps to solve problems where the geometric constraints involve the mutual tangency of two circles.

Transcript

00:01 In this problem of konig, we had given a matrix match type question and first is say here, the circle whose equation is x squared plus y square plus 6x plus k is equals to 0 and second circle is x square plus y square plus 8y minus 20 is equal to 0 are touching each other for the value of k.

00:27 So first there would be touching each other that means that means it would be internal and external touch so first we would determine the center so here we have center say you can say minus 3.

00:42 So minus 3 and y is 0.

00:47 Now similarly here this is the circle which is 0 axis 0 and y is say this is minus 4 and now we can find the radius also so r1 is say this would be under root of 9 minus k and from here we can say r2 is simply 6 now r 1 r2 so we have internal touch when r1 minus r2 is equals to c1 c2 c2 c2 is putting this value so this is equals to 5 so c1 c2 is 5 and r1 minus r2 so this would be under root of 9 minus k minus 6 and we have to take the modulars now when we solve for k so k is equals to 8 and minus 112 so we can match the option a with two options so this would be p and s as well as so p and s now we are working on b part b so part b says the short at distance from the point which is 2 comma minus 7 to the circle say circle equation is x square plus y square minus 14 x minus 10y minus 151 is equals to 0 is shorter distance that means perpendicular distance so we can determine the here so first we would determine the center so center is so this would be 7 and 5 so this is the center now we can find the radius so radius is also we can determine say this would be g square plus f square minus c so this would be under root of seven square which is 49 plus 5 square is 25 and plus 151 is equal to this would be say under root of 225 is equal to 15 now radius is 15 is 15 and we can find the distance between this point so we are naming at a so ac is equals to by distance formula so this would be 5 square and this would be here say under root of say 5 square is 25 and here this would be 12 square is 144 give you 13 so ac is equal to 13 and radius is r so minimum distance would be say we can say r minus ac so putting this value we are getting 2...

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