The coordinates of the focus of the parabola are

Explanation: To find the focus of the parabola, we need to convert the given general equation into the standard form . 1. Rearrange the Equation: Group the terms on one side and move the and constant terms to the right: Divide the entire equation by 4 to simplify the coefficient: 2. Complete the Square for : To complete the square for , add to both sides: 3. Factor out the Coefficient of : 4. Identify Standard Parameters: Comparing this with : Vertex : Value: 5. Calculate the Focus: For a parabola opening to the left, the focus is given by : To solve the -coordinate, find a common denominator: So, the focus is . Conclusion: The coordinates of the focus are . Correct Option: C)

How do I solve NIMCET 2022 Mathematics questions?

Practice NIMCET 2022 Mathematics previous year questions with step-by-step solutions on Mathem Solvex. Each question includes a detailed explanation and video solution to help you master the concept quickly.

What is the difficulty level of NIMCET Mathematics questions?

NIMCET Mathematics questions range from moderate to advanced difficulty. Consistent practice with official previous year papers and topic-wise drills on Mathem Solvex is the most effective preparation strategy.

Are NIMCET previous year papers available for free?

Yes. Mathem Solvex provides free access to NIMCET previous year question papers and topic-wise PDFs. You can download them or attempt them as timed live mock tests directly on the platform.

Tip:A–D to answerE for explanationV for videoS to reveal answer

NIMCET 2022 — Mathematics PYQ

NIMCET | Mathematics | 2022

The coordinates of the focus of the parabola $4 y^{2} + 12 x - 20 y + 67 = 0$ are

Choose the correct answer:

B.
$(- \frac{17}{2}, \frac{5}{4})$

C.
$(- \frac{17}{4}, \frac{5}{2})$
(Correct Answer)

D.
$(- \frac{5}{2}, \frac{17}{4})$

Explanation

To find the focus of the parabola, we need to convert the given general equation into the standard form $(y - k)^2 = 4a(x - h)$ .

1. Rearrange the Equation:

Group the $y$ terms on one side and move the $x$ and constant terms to the right:

$4y^2 - 20y = -12x - 67$

Divide the entire equation by 4 to simplify the $y^2$ coefficient:

$y^2 - 5y = -3x - \frac{67}{4}$

2. Complete the Square for $y$ :

To complete the square for $y^2 - 5y$ , add $(\frac{5}{2})^2 = \frac{25}{4}$ to both sides:

$y^2 - 5y + \frac{25}{4} = -3x - \frac{67}{4} + \frac{25}{4}$

$(y - \frac{5}{2})^2 = -3x - \frac{42}{4}$

$(y - \frac{5}{2})^2 = -3x - \frac{21}{2}$

3. Factor out the Coefficient of $x$ :

$(y - \frac{5}{2})^2 = -3(x + \frac{21}{6})$

$(y - \frac{5}{2})^2 = -3(x + \frac{7}{2})$

4. Identify Standard Parameters:

Comparing this with $(y - k)^2 = -4a(x - h)$ :

Vertex $(h, k)$ : $(-\frac{7}{2}, \frac{5}{2})$

$4a$ Value: $4a = 3 \implies a = \frac{3}{4}$

5. Calculate the Focus:

For a parabola opening to the left, the focus is given by $(h - a, k)$ :

$Focus = (-\frac{7}{2} - \frac{3}{4}, \frac{5}{2})$

To solve the $x$ -coordinate, find a common denominator:

$x = -\frac{14}{4} - \frac{3}{4} = -\frac{17}{4}$

So, the focus is $(-\frac{17}{4}, \frac{5}{2})$ .

Conclusion:

The coordinates of the focus are $(-\frac{17}{4}, \frac{5}{2})$ .

Correct Option:

C) $(-\frac{17}{4}, \frac{5}{2})$