Answer
Verified
64.8k+ views
Hint: First consider the coordinate of the point P. Then use the distance formula to obtain the distance between the point P and A, and that point P and B. Then substitute the values of PA and PB in the given equation to obtain the required locus.
Formula Used:
The distance formula of two points $(a,b),(c,d)$ is
$\sqrt {{{(c - a)}^2} + {{(d - b)}^2}} $.
Complete step by step solution:
Suppose that the coordinate of the point is $P(x,y)$ .
It is given that $P{A^2} - P{B^2} = 2{k^2}$--(1)
Now, $PA = \sqrt {{{(a - x)}^2} + {{(0 - y)}^2}} $
$P{A^2} = {(a - x)^2} + {y^2}$
And, $PB = \sqrt {{{( - a - x)}^2} + {{(0 - y)}^2}} $
$P{B^2} = {(a + x)^2} + {y^2}$
Now, from the equation (1) we have,
${(a - x)^2} + {y^2} - \left[ {{{(a + x)}^2} + {y^2}} \right] = 2{k^2}$
$\Rightarrow {a^2} - 2ax + {x^2} + {y^2} - \left[ {{a^2} + 2ax + {x^2} + {y^2}} \right] = 2{k^2}$
$\Rightarrow - 4ax = 2{k^2}$
$\Rightarrow 2ax + {k^2} = 0$
Option ‘A’ is correct
Note: Always consider the coordinates of P other than (h, k) to avoid confusion between the constant k and the coordinate k. Here using the distance formula of two points $(a,b),(c,d)$ it becomes easy to equate the given coordinates. If the diameter line is also bisecting the sides and is perpendicular then we have to find all equations of sides and equations of circle and then the coordinates of vertices.
Formula Used:
The distance formula of two points $(a,b),(c,d)$ is
$\sqrt {{{(c - a)}^2} + {{(d - b)}^2}} $.
Complete step by step solution:
Suppose that the coordinate of the point is $P(x,y)$ .
It is given that $P{A^2} - P{B^2} = 2{k^2}$--(1)
Now, $PA = \sqrt {{{(a - x)}^2} + {{(0 - y)}^2}} $
$P{A^2} = {(a - x)^2} + {y^2}$
And, $PB = \sqrt {{{( - a - x)}^2} + {{(0 - y)}^2}} $
$P{B^2} = {(a + x)^2} + {y^2}$
Now, from the equation (1) we have,
${(a - x)^2} + {y^2} - \left[ {{{(a + x)}^2} + {y^2}} \right] = 2{k^2}$
$\Rightarrow {a^2} - 2ax + {x^2} + {y^2} - \left[ {{a^2} + 2ax + {x^2} + {y^2}} \right] = 2{k^2}$
$\Rightarrow - 4ax = 2{k^2}$
$\Rightarrow 2ax + {k^2} = 0$
Option ‘A’ is correct
Note: Always consider the coordinates of P other than (h, k) to avoid confusion between the constant k and the coordinate k. Here using the distance formula of two points $(a,b),(c,d)$ it becomes easy to equate the given coordinates. If the diameter line is also bisecting the sides and is perpendicular then we have to find all equations of sides and equations of circle and then the coordinates of vertices.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
Electric field due to uniformly charged sphere class 12 physics JEE_Main
According to classical free electron theory A There class 11 physics JEE_Main
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main