Here you will learn how to find the focus of parabola with examples.
Letโs begin โ
Focus of Parabola Coordinates
(i) For Parabola \(y^2\) = 4ax :
The coordinates of focus is (a, 0).
(ii) For Parabola \(y^2\) = -4ax :
The coordinates of focus is (-a, 0).
(iii) For Parabola \(x^2\) = 4ay :
The coordinates of focus is (0, a).
(iv) For Parabola \(x^2\) = -4ay :
The coordinates of focus is (0, -a).
(v) For Parabola \((y โ k)^2\) = 4a(x โ h) :
The coordinates of focus is (h + a, k).
(vi) For Parabola \((x โ p)^2\) = 4a(y โ q) :
The coordinates of focus is (p, a + q).
Also Read : Different Types of Parabola Equations
Example : For the given parabola, find the coordinates of the foci :
(i) \(y^2\) = 8x
(ii) \(x^2\) = -16y
Solution :
(i) The given parabola is of the form \(y^2\) = 4ax, where 4a = 8 i.e. a = 2.
Hence, the coordinates of the focus are (a, 0) i.e. (2, 0).
(ii) The given parabola is of the form \(x^2\) = -4ay, where 4a = 16 i.e. a = 4.
Hence, the coordinates of the focus are (0, -a) i.e. (0, -4).
Example : Find the coordinates of foci of the parabola \(y^2 โ 8y โ x + 19\) = 0
Solution : The given equation is
\(y^2 โ 8y โ x + 19\) = 0ย \(\implies\)ย \(y^2 โ 8y\) = x โ 19
\(\implies\)ย \(y^2 โ 8y + 16\) = x โ 19 + 16
\(\implies\)ย \((y โ 4)^2\) = (x โ 3)
The equation is of the form \((y โ k)^2\) = 4a(x โ h),
On Comparing we get,
4a = 1 i.e. a = 1/4 and k = 4, h = 3
Hence, the coordinates of focus is (h + a, k).
i.e.ย ((3 + 1/4), 4) = (13/4, 4).
