JEE Main 2020 (9 January Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Circles: If the curves $x^{2} - 6 x + y^{2} + 8 = 0$ and $x^{2} - 8 y + y^{2} + 16 - k = 0, (k > 0)$ touch each other at a point, then the largest value of $k$ is _________.

Q: Why is there an absolute value in the internal touch condition?

The condition for internal contact is $d = |r_1 - r_2|$. We use the absolute value because we don't know *a priori* which circle is larger. If $r_2 > r_1$, the difference is $r_2 - r_1$; if $r_1 > r_2$, it is $r_1 - r_2$. The absolute value covers both scenarios, ensuring we don't miss any potential solutions.

Q: Can $k$ be negative?

The problem explicitly states $k > 0$. Furthermore, the radius of the second circle is $\sqrt{k}$. If $k$ were negative, the radius would be imaginary, which is physically impossible in the context of the coordinate plane. Thus, we only consider positive values of $k$.

Enter Numerical Value:

Visualized Solution

Problem Setup: Two Touching Circles

Given curves: $x^{2} - 6 x + y^{2} + 8 = 0$
And: $x^{2} - 8 y + y^{2} + 16 - k = 0, (k > 0)$
Goal: Find the largest value of $k$ if they touch.

Analyze Circle 1: $C_{1}$

Focus on Circle 1: $x^{2} - 6 x + y^{2} + 8 = 0$
We need to convert this to the standard form: $(x - h)^{2} + (y - k)^{2} = r^{2}$

Standard Form of $C_{1}$

Complete the square for $x$ : $(x^{2} - 6 x + 9) - 9 + y^{2} + 8 = 0$
Simplifies to: $(x - 3)^{2} + y^{2} = 1$
Center $C_{1} = (3, 0)$ , Radius $r_{1} = 1$

Analyze Circle 2: $C_{2}$

Focus on Circle 2: $x^{2} + y^{2} - 8 y + 16 - k = 0$
Group $y$ terms: $x^{2} + (y^{2} - 8 y + 16) = k$

Standard Form of $C_{2}$

Notice $(y^{2} - 8 y + 16)$ is a perfect square.
Simplifies to: $x^{2} + (y - 4)^{2} = k$
Center $C_{2} = (0, 4)$ , Radius $r_{2} = k$

Distance Between Centers $d$

The distance $d$ between centers $C_{1} (3, 0)$ and $C_{2} (0, 4)$ is crucial.
Distance formula: $d = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}$

Calculate $d$

Substitute coordinates: $d = (3 - 0)^{2} + (0 - 4)^{2}$
$d = 9 + 16 = 25$
$d = 5$

Conditions for Touching

Two circles touch if they satisfy either:
External Touch: $d = r_{1} + r_{2}$
Internal Touch: $d = ∣ r_{1} - r_{2} ∣$

Case 1: External Touch

Case 1 (External Touch): $d = r_{1} + r_{2}$
Substitute known values: $5 = 1 + k$

Solve for $k$ (External)

Solve for $k$ :
$k = 4$
Square both sides: $k = 16$

Case 2: Internal Touch

Case 2 (Internal Touch): $d = ∣ r_{1} - r_{2} ∣$
Substitute known values: $5 = ∣1 - k ∣$

Solve for $k$ (Internal)

Solve the absolute value equation:
Possibility A: $1 - k = 5 \Rightarrow k = - 4$ (Rejected)
Possibility B: $1 - k = - 5 \Rightarrow k = 6$
Square both sides: $k = 36$

Final Conclusion: Largest $k$

Possible values for $k$ are $16$ and $36$ .
The largest value of $k$ is $36$ .
Key Takeaway: Always check the internal touch condition $∣ r_{1} - r_{2} ∣$ !

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The Sigma Insight: Standard and General Equation of a Circle

Solution Diagram

Analyzing the First Circle

The first curve is given by the equation

x^{2} - 6 x + y^{2} + 8 = 0

. To understand its geometry, we complete the square for the

x

terms:

(x^{2} - 6 x + 9) - 9 + y^{2} + 8 = 0

This simplifies to the standard form:

(x - 3)^{2} + y^{2} = 1

This reveals a circle with center

C_{1} = (3, 0)

and radius

r_{1} = 1

Analyzing the Second Circle

Now, we turn our attention to the second curve:

x^{2} - 8 y + y^{2} + 16 - k = 0

. Completing the square for the

y

terms, we recognize that

y^{2} - 8 y + 16

is a perfect square:

x^{2} + (y - 4)^{2} = k

This circle is anchored at center

C_{2} = (0, 4)

, and its radius is

r_{2} = k

The Distance Between Centers

The most vital piece of information in any problem involving touching circles is the distance

d

between their centers. Using the distance formula for

C_{1} (3, 0)

and

C_{2} (0, 4)

d = (3 - 0)^{2} + (0 - 4)^{2} = 3^{2} + (- 4)^{2} = 9 + 16 = 5

No matter how

k

changes, the centers remain fixed at a distance of

5

units apart.

The Two Ways to Touch

Two circles can touch in two distinct ways: externally, where they sit side-by-side, or internally, where one is nestled inside the other.

For external contact, the distance between centers must equal the sum of the radii:

d = r_{1} + r_{2}

5 = 1 + k

Solving for

k

, we find

k = 4

, which leads to $k = 16$ .

For internal contact, the distance between centers must equal the absolute difference of the radii:

d = ∣ r_{1} - r_{2} ∣

5 = ∣1 - k ∣

This absolute value equation branches into two possibilities:

1 - k = 5 \Rightarrow k = - 4

(Rejected, as radius cannot be negative). 2.

1 - k = - 5 \Rightarrow k = 6

Squaring both sides of the second case, we find $k = 36$ .

Final Conclusion

We have identified two possible values for

k

16

and

36

. Since the question asks for the largest value, the final answer is:

$k = 36$

Common Traps & FAQs

Wait, why do we need to complete the square?

Completing the square is the standard algebraic technique to convert a general second-degree equation of a circle into its center-radius form. Without this, we cannot easily identify the center

(h, k)

and the radius

r

, which are essential for calculating the distance between centers and comparing it to the radii.

Why is there an absolute value in the internal touch condition?

The condition for internal contact is

d = ∣ r_{1} - r_{2} ∣

. We use the absolute value because we don't know *a priori* which circle is larger. If

r_{2} > r_{1}

, the difference is

r_{2} - r_{1}

; if

r_{1} > r_{2}

, it is

r_{1} - r_{2}

. The absolute value covers both scenarios, ensuring we don't miss any potential solutions.

Can

k

be negative?

The problem explicitly states

k > 0

. Furthermore, the radius of the second circle is

k

. If

k

were negative, the radius would be imaginary, which is physically impossible in the context of the coordinate plane. Thus, we only consider positive values of

k

How do I know if it's internal or external?

You don't have to know! That is the beauty of the math. By solving for both

d = r_{1} + r_{2}

and

d = ∣ r_{1} - r_{2} ∣

, you mathematically account for both possibilities. The geometry will naturally yield the correct values for

k

that satisfy either condition.

Animated Solution for Mathematics - Circles: If the curves $x^{2} - 6 x + y^{2} + 8 = 0$ and $x^{2} - 8 y + y^{2} + 16 - k = 0, (k > 0)$ touch each other at a point, then the largest value of $k$ is _________.

Enter Numerical Value:

Visualized Solution

Problem Setup: Two Touching Circles

Analyze Circle 1: $C_{1}$

Standard Form of $C_{1}$

Analyze Circle 2: $C_{2}$

Standard Form of $C_{2}$

Distance Between Centers $d$

Calculate $d$

Conditions for Touching

Case 1: External Touch

Solve for $k$ (External)

Case 2: Internal Touch

Solve for $k$ (Internal)

Final Conclusion: Largest $k$

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the First Circle

Analyzing the Second Circle

The Distance Between Centers

The Two Ways to Touch

Final Conclusion

Similar Questions

If the curves, $x^{2} - 6 x + y^{2} + 8 = 0$ and $x^{2} - 8 y + y^{2} + 16 - k = 0$ , $(k > 0)$ touch each other at a point, then the largest value of $k$ is

If the circles $x^{2} + y^{2} + 6 x + 8 y + 16 = 0$ and $x^{2} + y^{2} + 2 (3 - 3) x + 2 (4 - 6) y = k + 63 + 86, k > 0$ , touch internally at the point $P (α, β)$ , then $(α + 3)^{2} + (β + 6)^{2}$ is equal to ________

If the two circles $(x - 1)^{2} + (y - 3)^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct point, then

Let $C : x^{2} + y^{2} = 4$ and $C^{'} : x^{2} + y^{2} - 4 λ x + 9 = 0$ be two circles. If the set of all values of $λ$ so that the circles $C$ and $C^{'}$ intersect at two distinct points, is $R - [a, b]$ , then the point $(8 a + 12, 16 b - 20)$ lies on the curve :

If the two circles $(x - 1)^{2} + (y - 3)^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct points, then

The circles $x^{2} + y^{2} - 10 x + 16 = 0$ and $x^{2} + y^{2} = r^{2}$ intersect each other in two distinct points if

The circle $x^{2} + y^{2} - 4 x - 4 y + 4 = 0$ is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is $x + y - x y + k (x^{2} + y^{2})^{1/2} = 0$ . Find $k$ .

The minimum distance between any two points $P_{1}$ and $P_{2}$ while considering point $P_{1}$ on one circle and point $P_{2}$ on the other circle for the given circles' equations $x^{2} + y^{2} - 10 x - 10 y + 41 = 0$ and $x^{2} + y^{2} - 24 x - 10 y + 160 = 0$ is ____.

Two parallel chords of a circle of radius 2 are at a distance $3 + 1$ apart. If the chords subtend at the center, angles of $π / k$ and $2 π / k$ , where $k > 0$ , then the value of $[k]$ is ____.

The two circles $x^{2} + y^{2} = a x$ and $x^{2} + y^{2} = c^{2}$ ( $c > 0$ ) touch each other if

Animated Solution for Mathematics - Circles: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the curves x2−6x+y2+8=0 and x2−8y+y2+16−k=0,(k>0) touch each other at a point, then the largest value of k is _________.

Enter Numerical Value:

Visualized Solution

Problem Setup: Two Touching Circles

Analyze Circle 1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C1​

Standard Form of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C1​

Analyze Circle 2: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Standard Form of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } C2​

Distance Between Centers .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d

Calculate .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } d

Conditions for Touching

Case 1: External Touch

Solve for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k (External)

Case 2: Internal Touch

Solve for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k (Internal)

Final Conclusion: Largest .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the First Circle

Analyzing the Second Circle

The Distance Between Centers

The Two Ways to Touch

Final Conclusion

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the curves, x2−6x+y2+8=0 and x2−8y+y2+16−k=0, (k>0) touch each other at a point, then the largest value of k is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the circles x2+y2+6x+8y+16=0 and x2+y2+2(3−3​)x+2(4−6​)y=k+63​+86​,k>0, touch internally at the point P(α,β), then (α+3​)2+(β+6​)2 is equal to ________

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the two circles (x−1)2+(y−3)2=r2 and x2+y2−8x+2y+8=0 intersect in two distinct point, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the two circles (x−1)2+(y−3)2=r2 and x2+y2−8x+2y+8=0 intersect in two distinct points, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The circles x2+y2−10x+16=0 and x2+y2=r2 intersect each other in two distinct points if

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The two circles x2+y2=ax and x2+y2=c2 (c>0) touch each other if

Animated Solution for Mathematics - Circles: If the curves $x^{2} - 6 x + y^{2} + 8 = 0$ and $x^{2} - 8 y + y^{2} + 16 - k = 0, (k > 0)$ touch each other at a point, then the largest value of $k$ is _________.

Analyze Circle 1: $C_{1}$

Standard Form of $C_{1}$

Analyze Circle 2: $C_{2}$

Standard Form of $C_{2}$

Distance Between Centers $d$

Calculate $d$

Solve for $k$ (External)

Solve for $k$ (Internal)

Final Conclusion: Largest $k$

If the curves, $x^{2} - 6 x + y^{2} + 8 = 0$ and $x^{2} - 8 y + y^{2} + 16 - k = 0$ , $(k > 0)$ touch each other at a point, then the largest value of $k$ is

If the circles $x^{2} + y^{2} + 6 x + 8 y + 16 = 0$ and $x^{2} + y^{2} + 2 (3 - 3) x + 2 (4 - 6) y = k + 63 + 86, k > 0$ , touch internally at the point $P (α, β)$ , then $(α + 3)^{2} + (β + 6)^{2}$ is equal to ________

If the two circles $(x - 1)^{2} + (y - 3)^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct point, then

If the two circles $(x - 1)^{2} + (y - 3)^{2} = r^{2}$ and $x^{2} + y^{2} - 8 x + 2 y + 8 = 0$ intersect in two distinct points, then

The circles $x^{2} + y^{2} - 10 x + 16 = 0$ and $x^{2} + y^{2} = r^{2}$ intersect each other in two distinct points if

The two circles $x^{2} + y^{2} = a x$ and $x^{2} + y^{2} = c^{2}$ ( $c > 0$ ) touch each other if