Question

The line $y = x + 5$ touches

• A. the parabola y2= 20x
• B. the ellipse 9x2+ 16y2= 144
• C. the hyperbola x2/ 29 - y2/ 4 = 1
• D. the circle x2+ y2 = 25

Answer 1

The Correct Option is A, B, C

We are given the line \( y = x + 5 \), and we need to determine which conic section it touches.

Step 1: The Parabola \( y^2 = 20x \)
For the parabola \( y^2 = 20x \), the equation can be rewritten as: \[ y = \pm \sqrt{20x} \] The line \( y = x + 5 \) will touch this parabola if it intersects it at exactly one point. Substituting \( y = x + 5 \) into the equation \( y^2 = 20x \): \[ (x + 5)^2 = 20x \] Expanding: \[ x^2 + 10x + 25 = 20x \] Simplifying: \[ x^2 - 10x + 25 = 0 \] The discriminant of this quadratic equation is: \[ \Delta = (-10)^2 - 4(1)(25) = 100 - 100 = 0 \] Since the discriminant is 0, the line touches the parabola at exactly one point. 

Step 2: The Ellipse \( 9x^2 + 16y^2 = 144 \)
Rewriting the equation of the ellipse: \[ \frac{x^2}{16} + \frac{y^2}{9} = 1 \] Substituting \( y = x + 5 \) into this equation: \[ \frac{x^2}{16} + \frac{(x + 5)^2}{9} = 1 \] This gives a quadratic equation that will have two solutions, meaning the line intersects the ellipse at two points. Therefore, the line does not touch the ellipse. 

Step 3: The Hyperbola \( \frac{x^2}{29} - \frac{y^2}{4} = 1 \)
Rewriting the equation of the hyperbola: \[ \frac{x^2}{29} - \frac{y^2}{4} = 1 \] Substituting \( y = x + 5 \): \[ \frac{x^2}{29} - \frac{(x + 5)^2}{4} = 1 \] This gives a quadratic equation that will have two solutions, meaning the line intersects the hyperbola at two points. Therefore, the line does not touch the hyperbola. 

Step 4: The Circle \( x^2 + y^2 = 25 \)
Rewriting the equation of the circle: \[ x^2 + y^2 = 25 \] Substituting \( y = x + 5 \): \[ x^2 + (x + 5)^2 = 25 \] This gives a quadratic equation that will have two solutions, meaning the line intersects the circle at two points. Therefore, the line does not touch the circle.

Answer:

\[ \boxed{\text{The parabola } y^2 = 20x} \]