Question

Is the tangent at (1,7) to the curre x²=y-6 touches
the circle x2+y2+16x+12y+c=0 , then the value of. C is
(a) 95
(6) l95 (C) 185 (d) 85​

Answer 1

EXPLANATION.

Tangent at (1,7) to the curve x² = y - 6.

Touches the circle x² + y² + 16x + 12y + c = 0.

A we know that,

General equation of circle.

⇒ x² + y² + 2gx + 2fy + c = 0.

Centre of circle = (-g,-f).

Centre of circle = (-8,-6).

Radius of circle = √g² + f² - c.

Radius of circle = √(-8)² + (-6)² - c.

Radius of circle = √64 + 36 - c.

Now, we need to find slope of the equation, we get.

⇒ x² = y - 6.

Differentiate w.r.t x, we get.

⇒ 2x = dy/dx - 0..

⇒ 2x = dy/dx.

Put the value of x = 1 in the equation, we get.

⇒ 2(1) = dy/dx.

⇒ dy/dx = 2.

dy/dx = slope of tangent.

As we know that,

Equation of tangent.

⇒ (y - y₁) = m(x - x₁).

⇒ (y - 7) = 2(x - 1).

⇒ y - 7 = 2x - 2.

⇒ 2x - 2 - y + 7 = 0.

⇒ 2x - y + 5 = 0.

As we know that,

Length of perpendicular.

From (x₁, y₁) to a straight line ax + by + c = 0 then,

⇒ P = |ax₁ + by₁ + c/√a² + b²|.

Perpendicular = Radius.

⇒ |2x - y + 5/√(2)² + (1)²| = √64 + 36 - c.

⇒ |2(-8) - (-6) + 5/√4 + 1 | = √100 - c.

⇒ | - 16 + 6 + 5/√5 | = √100 - c.

⇒ | - 5/√5| = √100 - c.

⇒ √5 = √100 - c.

⇒ 5 = 100 - c.

⇒ 5 - 100 = - c.

⇒ - 95 = - c.

⇒ c = 95.

Option [A] is correct answer.

Answer 2

Step-by-step explanation:

Given equation of curve is

$\tt \: {x}^{2} = y - 6$

$\implies \tt \: y = x + 6 \\ \implies \tt \: \frac{dy}{dx} = 2x$

Slope of tangent is m

$\tt \: = | \frac{dy}{dx} |_{(0,7) } = 2 \times 1 = 2$

Equation of tangent at the point(1,7) is given by

$\tt \: y - 7 = 2(x - 1) \\ \tt \longrightarrow \: y - 7 = 2x - 2 \\ \tt \longrightarrow \: 2x - y + 5 = 0$

Given equation of circle is

$\tt \longrightarrow \: {x}^{2} + {y }^{2} + 16x + 12y + c = 0 \\ \tt \longrightarrow {(x + 8)}^{2} + {(y + 6)}^{2} + c - 64 - 36 = 0 \\ \tt \longrightarrow{(x + 8)}^{2} + {(y + 6)}^{2} = 100 - c$

If the tangent touches the circle , then distance from the centre =radius

= Distance of (-8,-6) from 2x-y-5=0 is the radius

$\tt \longrightarrow\: | \frac{2( - 8) - ( - 6)}{ \sqrt{4 + 1} } | = | \sqrt{5} | \\ \tt \longrightarrow \: Radius = \sqrt{100 - c} = \sqrt{5} \\ \tt \longrightarrow \: c = \blue{95}$