Question:
The equation of the plane that passes through the points \( (1, 0, 2), (-1, 1, 2) \) and \( (5, 0, 3) \) is:
The equation of the plane that passes through the points \( (1, 0, 2), (-1, 1, 2) \) and \( (5, 0, 3) \) is:
Show Hint
In a multiple-choice format, it is often faster to plug in a single point. For example, point (5, 0, 3) in option (A) gives \( 5 + 0 - 12 + 7 = 0 \), which is true. Checking the other options with the same point will quickly rule them out.
Updated On: May 1, 2026
- \( x + 2y - 4z + 7 = 0 \)
- \( x + 2y - 3z + 7 = 0 \)
- \( x - 2y + 4z + 7 = 0 \)
- \( 2y - 4z - 7 + x = 0 \)
- \( x + 2y + 3z + 7 = 0 \)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept: The general equation of a plane passing through three points \( (x_1, y_1, z_1), (x_2, y_2, z_2), \) and \( (x_3, y_3, z_3) \) can be found using the determinant: \[ \begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \end{vmatrix} = 0 \] Alternatively, you can test the given points in each option to find which one satisfies all three.
Step 1: Substitute points into the determinant form.
Using \( P_1(1, 0, 2) \): \[ \begin{vmatrix} x-1 & y-0 & z-2 \\ -1-1 & 1-0 & 2-2 \\ 5-1 & 0-0 & 3-2 \end{vmatrix} = 0 \implies \begin{vmatrix} x-1 & y & z-2 \\ -2 & 1 & 0 \\ 4 & 0 & 1 \end{vmatrix} = 0 \]
Step 2: Expand the determinant to find the equation.
Expanding along the first row: \[ (x-1)(1 - 0) - y(-2 - 0) + (z-2)(0 - 4) = 0 \] \[ (x-1) + 2y - 4(z-2) = 0 \] \[ x - 1 + 2y - 4z + 8 = 0 \] \[ x + 2y - 4z + 7 = 0 \]
Was this answer helpful?
0
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 Plane Questions
- The equation of the plane containing the line \( \frac{x-\alpha}{l} = \frac{y-\beta}{m} = \frac{z-\gamma}{n} \) is \( a(x-\alpha)+b(y-\beta)+c(z-\gamma)=0 \), where \( al + bm + cn \) is equal to
- A plane passes through the point \( (1,-2,1) \) and is perpendicular to planes \(2x-2y+z=0\) and \(x-y+2z=4\). Then the equation of the plane is
- The equation of the plane containing the lines \[ \frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7} \text{and} \frac{x-2}{2} = \frac{y-4}{3} = \frac{z-6}{5} \]
- A plane passes through the point $(0,1,1)$ and has normal vector $\hat{i}+\hat{j}+\hat{k}$. Its equation is
- The sum of the intercepts made by the plane $\vec{r}\cdot(3\hat{i}+\hat{j}+2\hat{k})=18$ on the co-ordinate axes 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