Question:
If the line \(\frac{x}{a} + \frac{y}{b} = 1\) moves in such a way that \(\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}\) where \(c\) is a constant, then the locus of the foot of perpendicular from the origin on the straight line is
If the line \(\frac{x}{a} + \frac{y}{b} = 1\) moves in such a way that \(\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2}\) where \(c\) is a constant, then the locus of the foot of perpendicular from the origin on the straight line is
Show Hint
Foot of perpendicular from origin to \(\frac{x}{a}+\frac{y}{b}=1\) is \(\left(\frac{ab^2}{a^2+b^2}, \frac{a^2b}{a^2+b^2}\right)\).
Updated On: Apr 23, 2026
- Straight line
- Parabola
- Ellipse
- Circle
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Formula / Definition}
\[ \text{Foot of perpendicular } (h,k): h = \frac{ab^2}{a^2+b^2}, k = \frac{a^2b}{a^2+b^2} \]
Step 2: Calculation / Simplification}
\[ h^2 + k^2 = \frac{a^2b^4 + a^4b^2}{(a^2+b^2)^2} = \frac{a^2b^2(a^2+b^2)}{(a^2+b^2)^2} = \frac{a^2b^2}{a^2+b^2} \]
Given \(\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2} \Rightarrow \frac{a^2+b^2}{a^2b^2} = \frac{1}{c^2}\)
\(\frac{a^2b^2}{a^2+b^2} = c^2\)
\(\therefore h^2 + k^2 = c^2\)
Step 3: Final Answer
\[ x^2 + y^2 = c^2 \text{ (Circle)} \]
\[ \text{Foot of perpendicular } (h,k): h = \frac{ab^2}{a^2+b^2}, k = \frac{a^2b}{a^2+b^2} \]
Step 2: Calculation / Simplification}
\[ h^2 + k^2 = \frac{a^2b^4 + a^4b^2}{(a^2+b^2)^2} = \frac{a^2b^2(a^2+b^2)}{(a^2+b^2)^2} = \frac{a^2b^2}{a^2+b^2} \]
Given \(\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{c^2} \Rightarrow \frac{a^2+b^2}{a^2b^2} = \frac{1}{c^2}\)
\(\frac{a^2b^2}{a^2+b^2} = c^2\)
\(\therefore h^2 + k^2 = c^2\)
Step 3: Final Answer
\[ x^2 + y^2 = c^2 \text{ (Circle)} \]
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Number Theory Questions
- If $(3 - x) \equiv (2x - 5) \pmod{4}$, then one of the values of $x$ is
- The least positive remainder when $123 \times 125 \times 127$ is divided by 124 is
- If $a$, $b$, $c$ are different integers such that $(a,b) = c$, which of the following statements is true?
- If $77 \equiv 88x \pmod{5}$, then x is equal to
- When $5^{20}$ is divided by 7, the remainder is
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions