Equation of the common tangent, with positive slope, to the circle x² + y² - 8x = 0 as well as to the hyperbola , is:

Concept:

• The equation of a line, with slope m, is: y = mx + c.

• The distance between a point P(x₁, y₁) and the line ax + by + c = 0 is given by: Distance = .

• The equation of a circle with center at O(a, b) and radius r, is given by: (x - a)² + (y - b)² = r².
• Tangent to a Hyperbola: If the line y = mx + c touches the hyperbola , then c² = a²m² - b². The equation of the tangent is: . Either of the lines is the equation of the tangent but not both.

Calculation:

The equation of the circle can be written as (x - 4)2 + y2 = 42.

Comparing with the general form of a circle, we have center O(4, 0) and radius r = 4.

The equation of the given hyperbola can be written as .

Comparing with the general form of a hyperbola, we have a = 3 and b = 2.

The equation of the tangent to this hyperbola will have the form:

⇒ 

Since this line is a tangent to the circle as well, we must have:

Distance from the center O(4, 0) of the circle to the tangent  = radius (r = 4) of the circle.

Using the formula for the distance of a point from a line, we get:

⇒ 

On squaring both sides, we get:

⇒ 

⇒ 

Squaring again, we get:

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

⇒ 

Discarding the negative value of m²:

⇒ 

Since the slope is given to be positive, we get:

⇒ 

∴ Equation of the tangent will be:

⇒ 

⇒ 

⇒ .

Additional Information

• The slope (m) of the tangent at a point P(a, b) to a curve y = f(x), is given by: .
• Tangent to a Parabola: The equation of the tangent to the parabola y2 = 4ax, at a point (x1, y1), is given by: yy1 = 2a(x + x1).

  Normal to a Parabola: The equation of the normal to the parabola y2 = 4ax, at a point (x1, y1), is given by: 2a(y - y1) = (-y1)(x - x1).

• Tangent to a Circle: The equation of the tangent to the circle x² + y² = r² at a point (x₁, y₁), is given by: xx₁ + yy₁ = r².

  Normal to a Circle: The equation of a normal to the circle x² + y² = r² at a point (x₁, y₁), is given by: yx₁ - xy₁ = 0.

• Tangent to an Ellipse: The equation of the tangent to the ellipse , at a point (x₁, y₁), is given by: .

  Normal to an Ellipse: The equation of the normal to the ellipse , at a point (x₁, y₁), is given by: .

• Tangent to a Hyperbola: The equation of the tangent to the hyperbola , at a point (x₁, y₁), is given by: .

  Normal to a Hyperbola: The equation of the normal to the hyperbola , at a point (x₁, y₁), is given by: .