Question:
If the equation of the plane passing through the point $A(-2, 1, 3)$ and perpendicular to the vector $3\vec{i} + \vec{j} + 5\vec{k}$ is $ax + by + cz + d = 0$, then $\dfrac{a + b}{c + d} =$
If the equation of the plane passing through the point $A(-2, 1, 3)$ and perpendicular to the vector $3\vec{i} + \vec{j} + 5\vec{k}$ is $ax + by + cz + d = 0$, then $\dfrac{a + b}{c + d} =$
Show Hint
Use point-normal form and substitute coordinates directly to get the plane equation.
Updated On: May 19, 2025
- $4/5$
- $2/3$
- $1$
- $-4/5$
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Normal vector to the plane is $\vec{n} = \langle 3, 1, 5 \rangle$
So plane equation: $3(x + 2) + 1(y - 1) + 5(z - 3) = 0$
$3x + 6 + y - 1 + 5z - 15 = 0 \Rightarrow 3x + y + 5z - 10 = 0$
Thus, $a = 3$, $b = 1$, $c = 5$, $d = -10$
Then: $\dfrac{a + b}{c + d} = \dfrac{3 + 1}{5 - 10} = \dfrac{4}{-5} = -\dfrac{4}{5}$
So plane equation: $3(x + 2) + 1(y - 1) + 5(z - 3) = 0$
$3x + 6 + y - 1 + 5z - 15 = 0 \Rightarrow 3x + y + 5z - 10 = 0$
Thus, $a = 3$, $b = 1$, $c = 5$, $d = -10$
Then: $\dfrac{a + b}{c + d} = \dfrac{3 + 1}{5 - 10} = \dfrac{4}{-5} = -\dfrac{4}{5}$
Was this answer helpful?
0
0
Top Questions on Vectors
- The diagonals of a parallelogram are given by \( \mathbf{a} = 2 \hat{i} - \hat{j} + \hat{k} \) and \( \mathbf{b} = \hat{i} + 3 \hat{j} - \hat{k}\) . Find the area of the parallelogram.
- If \( \mathbf{a} \) and \( \mathbf{b} \) are position vectors of two points \( P \) and \( Q \) respectively, then find the position vector of a point \( R \) in \( QP \) produced such that \[ QR = \frac{3}{2} QP. \]
- If \( \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \), \( |\overrightarrow{a}| = \sqrt{37} \), \( |\overrightarrow{b}| = 3 \), and \( |\overrightarrow{c}| = 4 \), then the angle between \( \overrightarrow{b} \) and \( \overrightarrow{c} \) is:
- If $\mathbf{a}$ and $\mathbf{b}$ are position vectors of point A and point B, respectively, find the position vector of point C on $\overrightarrow{BA}$ such that $BC = 3BA$.
If vector \( \mathbf{a} = 3 \hat{i} + 2 \hat{j} - \hat{k} \) \text{ and } \( \mathbf{b} = \hat{i} - \hat{j} + \hat{k} \), then which of the following is correct?
View More Questions
Questions Asked in AP EAPCET exam
- The number of all five-letter words (with or without meaning) having at least one repeated letter that can be formed by using the letters of the word INCONVENIENCE is:
- AP EAPCET - 2025
- Binomial Expansion
- If \(\alpha, \beta, \gamma\) are the roots of the equation \[ x^3 - 13x^2 + kx + 189 = 0 \] such that \(\beta - \gamma = 2\), then find the ratio \(\beta + \gamma : k + \alpha\).
- In a container of volume 16.62 m$^3$ at 0°C temperature, 2 moles of oxygen, 5 moles of nitrogen and 3 moles of hydrogen are present, then the pressure in the container is (Universal gas constant = 8.31 J/mol K)
- AP EAPCET - 2025
- Ideal gas equation
- If \[ A = \begin{bmatrix} x & 2 & 1 \\ -2 & y & 0 \\ 2 & 0 & -1 \end{bmatrix}, \] where \( x \) and \( y \) are non-zero real numbers, trace of \( A = 0 \), and determinant of \( A = -6 \), then the minor of the element 1 of \( A \) is:}
- AP EAPCET - 2025
- Complex numbers
- Two objects of masses 5 kg and 10 kg are placed 2 meters apart. What is the gravitational force between them?
(Use \(G = 6.67 \times 10^{-11}\, \mathrm{Nm^2/kg^2}\))- AP EAPCET - 2025
- Gravitation
View More Questions