Question

The tangent at \( (1, 7) \) to the curve \( x^2 = y - 6 \) touches the circle \( x^2 + y^2 + 16x + 12y + c = 0 \) at:

• A. \( (6, 7) \)
• B. \( (-6, 7) \)
• C. \( (6, -7) \)
• D. \( (-6, -7) \)

Hint:

When a tangent touches a circle at exactly one point, the discriminant of the corresponding quadratic equation must be zero.

Answer 1

The Correct Option is D

We are given two equations: 1. The equation of the curve: \( x^2 = y - 6 \), or equivalently \( y = x^2 + 6 \). 2. The equation of the circle: \( x^2 + y^2 + 16x + 12y + c = 0 \). The tangent at the point \( (1, 7) \) on the curve \( y = x^2 + 6 \) touches the circle at one point. Step 1: Finding the slope of the tangent to the curve.
The curve is \( y = x^2 + 6 \). To find the slope of the tangent at \( (1, 7) \), we differentiate \( y \) with respect to \( x \): \[ \frac{dy}{dx} = 2x. \] At \( x = 1 \), the slope of the tangent is: \[ \frac{dy}{dx} = 2(1) = 2. \] Thus, the equation of the tangent at \( (1, 7) \) is: \[ y - 7 = 2(x - 1), \] which simplifies to: \[ y = 2x + 5. \]
Step 2: Substituting the equation of the tangent into the equation of the circle.
We substitute \( y = 2x + 5 \) into the equation of the circle: \[ x^2 + (2x + 5)^2 + 16x + 12(2x + 5) + c = 0. \] Expanding the terms: \[ x^2 + (4x^2 + 20x + 25) + 16x + (24x + 60) + c = 0, \] \[ x^2 + 4x^2 + 20x + 25 + 16x + 24x + 60 + c = 0, \] \[ 5x^2 + 60x + 85 + c = 0. \] Now, the tangent touches the circle at exactly one point, so the discriminant of this quadratic equation in \( x \) must be zero.
Step 3: Setting the discriminant to zero.
The quadratic equation in \( x \) is: \[ 5x^2 + 60x + (85 + c) = 0. \] The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by: \[ \Delta = b^2 - 4ac. \] For the quadratic \( 5x^2 + 60x + (85 + c) = 0 \), we have: \[ a = 5, \quad b = 60, \quad c = 85 + c. \] The discriminant is: \[ \Delta = 60^2 - 4(5)(85 + c) = 3600 - 20(85 + c) = 3600 - 1700 - 20c = 1900 - 20c. \] For the tangent to touch the circle at exactly one point, the discriminant must be zero: \[ 1900 - 20c = 0, \] \[ 20c = 1900, \] \[ c = 95. \]
Step 4: Substituting \( c = 95 \) into the circle equation.
Substitute \( c = 95 \) back into the circle equation: \[ x^2 + y^2 + 16x + 12y + 95 = 0. \] Substitute \( y = 2x + 5 \) into this equation: \[ x^2 + (2x + 5)^2 + 16x + 12(2x + 5) + 95 = 0. \] Simplifying the above, we find the point where the tangent touches the circle. The point of tangency is \( (-6, -7) \).
Step 5: Conclusion.
Thus, the point of tangency is \( (-6, -7) \), and the correct answer is (d).