Question:
The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:
The focus of the parabola \(y^2 + 4y - 8x + 20 = 0\) is at the point:
Show Hint
When completing the square for a parabola, make sure to balance the equation by adding and subtracting the same values. Remember, the focus of a parabola \((y-k)^2 = 4p(x-h)\) lies \(p\) units from the vertex along the axis of symmetry.
Updated On: Apr 7, 2026
- \( (0, -2) \)
- \( (2, -2) \)
- \( (4, -2) \)
- \( (2, 0) \)
- \( (4, -4) \)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
First, rewrite the equation \(y^2 + 4y - 8x + 20 = 0\) in a form that reveals the vertex and direction:
\[
y^2 + 4y = 8x - 20
\]
Complete the square for the \(y\)-terms:
\[
(y+2)^2 - 4 = 8x - 20
\]
\[
(y+2)^2 = 8x - 16
\]
\[
(y+2)^2 = 8(x-2)
\]
This is a parabola that opens rightwards with the vertex form \((y-k)^2 = 4p(x-h)\), where \(k = -2\), \(h = 2\), and \(4p = 8\) so \(p = 2\).
The focus of a parabola \( (y-k)^2 = 4p(x-h) \) is at \( (h+p, k) \): \[ (h+p, k) = (2+2, -2) = (4, -2) \]
The focus of a parabola \( (y-k)^2 = 4p(x-h) \) is at \( (h+p, k) \): \[ (h+p, k) = (2+2, -2) = (4, -2) \]
Was this answer helpful?
1
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Magnitude and Directions of a Vector Questions
Let \(a=i+j+2k\) and \(b=i-2j+3k\) be two vectors. Then the unit vector in the direction of \(a-b\) is
- The direction cosines of the vector $\vec{a} = -2\hat{i} + \hat{j} - \hat{k}$ are
- The value of \(λ\) for which the vectors \(i+j-k\)and \(λi+3j+k\) are perpendicular is ?
- Let \(a\) and \(b\) be perpendicular vectors such that \(IaI=√104\) and \(IbI=6\).Then the value of\( Ia-bI\) is ?
- The position vectors of two points P and Q are given by \(OP=2a-b\) and \(OQ=a+3b\) ,respectively.If a point R divides the line joining P and Q internally in the ratio \(1:2\) ,then the position vector of the point R is ?
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions