Question

A. The line 2x + y = 5 is a diameter of the circle ${x^2} + {y^2} - 6x + 2y = 0$ R. Normal of a circle at any point always passes through the center of the circle.
 (a) Both A and R are individually true and R is the correct explanation of A
 (b) Both A and R are individually true but R is not the correct explanation of A
 (c) A is true but R is false
 (d) A is false but R is true

Answer 1

Hint:Centre lies on every line that is a diameter, which implies the intersection point of any two diameters of a given circle gives us the center. Normal is generally the perpendicular line to the tangent at a given point on the circle.

Complete step-by-step solution:
Given
The circle equation as ${x^2} + {y^2} - 6x + 2y = 0$
Therefore we can write this equation as
${x^2} + {y^2} - 6x + 2y + 9 + 1 - 10 = 0$
$\Rightarrow {x^2} - 6x + 9 + {y^2} + 2y + 1 = 10$ [$\because$ For making the given equation as perfect square we add and subtract the square of half of the coefficients of x and y]
$\Rightarrow {\left( {x - 3} \right)^2}$ + ${\left( {y + 1} \right)^2}$ = ${\left( {\sqrt {10} } \right)^2}$
Therefore this is in the form of ${\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {r^2}$ where the point (a,b) is the centre of the circle with radius r.

Therefore for the above circle the centre is $(3, -1)$ and radius of circle is $\sqrt {10} $
If a line is the diameter of the circle the line should pass through the center of the circle.
Therefore (3,$ - $ 1) should lie on the line $ 2x + y = 5.$
$ 2( 3 ) + ( -1) = 6 – 1 = 5$
Hence verified.
Therefore the line $2x + y = 5$ is the diameter of the given circle.
Let us assume a circle with center O and take a point P on the circle. From the point, P draw a tangent PA
Let us assume that a perpendicular through P passes through o’.
Therefore $\left| \!{\underline {\, {APO'} \,}} \right. $ = ${90^ \circ }$
Join O and P
We know that from the property of tangents the line drawn from the center to the point of contact of the tangent to the circle is always perpendicular.
Therefore OP is perpendicular to AP then
$\left| \!{\underline {\, {APO} \,}} \right. = {90^ \circ }$
From this, we can say that our assumption is wrong and we can say that the Normal of a circle at any point always passes through the center of the circle.
From this, we can say that Both A and R are individually true and R is the correct explanation of A, Option A is correct.

Note: Do not make calculation mistakes. Go through the formulae of the equation of circle and equation of tangent and equation of normal. read more concepts and make sure you are clear with the topic. And read the question carefully when it is about assertion and reasoning statements.