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If the tangent at (1,7) to the curve x^{2}=y-6 touches the cir

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Hard

Solving time: 4 mins

If the tangent at $(1, 7)$ to the curve $x^{2} = y - 6$ touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ , then the value of $c$ is

195

185

Views: 6,149 students

Updated on: Dec 14, 2023

Book:

Arihant JEE Main Chapterwise Solutions Mathematics (2019-2002) (Arihant)

Exam:

JEE Mains 2018

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Oliver Discussed

If the tangent at

(1, 7)

to the curve

x^{2} = y - 6

touches the circle

x^{2} + y^{2} + 16 x + 12 y + c = 0

, then the value of

c

7 mins ago

Discuss this question LIVE

7 mins ago

Text solutionVerified

Exp. (d)
Key Idea Equation of tangent to the curve

x^{2} = 4 a y

(x_{1}, y_{1})

x x_{1} = 4 a (\frac{y + y _{1}}{2})

Tangent to the curve

x^{2} = y - 6

(1, 7)

x = \frac{y + 7}{2} - 6

\Rightarrow 2 x - y + 5 = 0

.......(i)
Equation of circle is

x^{2} + y^{2} + 16 x + 12 y + c = 0

Centre

(- 8, - 6)

r = 8^{2} + 6^{2} - c = 100 - c

Since, line

2 x - y + 5 = 0

also touches the circle.

∴ 100 - c = ∣ ∣ \frac{2 ( - 8 ) - ( - 6 ) + 5}{2 ^{2} + 1 ^{2}} ∣ ∣

\Rightarrow 100 - c = ∣ ∣ \frac{- 16 + 6 + 5}{5} ∣ ∣

\Rightarrow 100 - c = ∣ - 5 ∣ \Rightarrow 100 - c = 5 \Rightarrow c = 95

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Question Text	If the tangent at $(1, 7)$ to the curve $x^{2} = y - 6$ touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ , then the value of $c$ is
Updated On	Dec 14, 2023
Topic	Conic Sections
Subject	Mathematics
Class	Class 11
Answer Type	Text solution:1 Video solution: 4
Upvotes	356
Avg. Video Duration	10 min

If the tangent at $(1, 7)$ to the curve $x^{2} = y - 6$ touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ , then the value of $c$ is

Text solutionVerified

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If the tangent at $(1, 7)$ to the curve $x^{2} = y - 6$ touches the circle $x^{2} + y^{2} + 16 x + 12 y + c = 0$ , then the value of $c$ is