Find the equation of the parabola with focus at \( (3,1) \) and vertex at \( (5,1) \).
Show Hint
- \( (y - 1)^2 = -8(x - 5) \)
- \( (y - 1)^2 = 8(x - 5) \)
- \( (y - 1)^2 = 8(x - 3) \)
- \( (y - 1)^2 = -8(x - 3) \)
- \( (y - 1)^2 = -4(x - 5) \)
The Correct Option is A
Solution and Explanation
The equation of a horizontal parabola is: \[ (y - k)^2 = 4p(x - h) \] where \( (h, k) \) is the vertex, and \( p \) is the distance from the vertex to the focus. Given: \[ (h, k) = (5,1), \quad (f_x, f_y) = (3,1) \] \[ p = f_x - h = 3 - 5 = -2 \] Substituting into the equation: \[ (y - 1)^2 = 4(-2)(x - 5) \] \[ (y - 1)^2 = -8(x - 5) \] Thus, the correct answer is:
Final Answer: \[ \boxed{(y - 1)^2 = -8(x - 5)} \]
Top Questions on Parabola
If \( x^2 = -16y \) is an equation of a parabola, then:
(A) Directrix is \( y = 4 \)
(B) Directrix is \( x = 4 \)
(C) Co-ordinates of focus are \( (0, -4) \)
(D) Co-ordinates of focus are \( (-4, 0) \)
(E) Length of latus rectum is 16
- Let P be the parabola, whose focus is $ (-2, 1) $ and directrix is $ 2x + y + 2 = 0 $. Then the sum of the ordinates of the points on P, whose abscissa is -2, is:
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Two parabolas have the same focus $(4, 3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:
- Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:
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